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MATHEMATICAL    MONOGRAPHS. 

tniTED    BV 

MANSFIELD   MERRIMAN  and  ROBERT   S.   WOODWARD. 


No.  7. 


PROBABILITY 


AND 


THEORY  OF  ERRORS. 


BY 


ROBERT    S.    WOODWARD, 

President  Carnegie  Institution  of  Washington. 


NEW  YORK: 

JOHN    WILEY    &    SONS. 

London:    CHAPMAN  &   HALL,    Limited. 


630526 


Copyright,  1896, 

liV 

MANSFIELD    MERRIMAN  and  ROBERT   S.   WOODWARD 

UNDER    THE    T    I  LE 

HIGHER    MATHEMATI-CS. 

First  Edilion,   September,  i8g6. 
Second   Edition,  January,  1898. 
Third  Edition,  August,  1900. 
Fourth  Edition,  January,  1906, 


4/22 


EDITORS'   PREFACE. 


The  volume  called  Higher  Mathematics,  the  first  edition 
of  which  was  published  in  1896,  contained  eleven  chapters  by 
eleven  authors,  each  chapter  being  independent  of  the  others, 
but  all  supposing  the  reader  to  have  at  least  a  mathematical 
training  equivalent  to  that  given  in  classical  and  engineering 
colleges.  The  pubhcation  of  that  volume  is  now  discontinued 
and  the  chapters  are  issued  in  separate  form.  In  these  reissues 
it  will  generally  be  found  that  the  monographs  are  enlarged 
by  additional  articles  or  appendices  which  either  amplify  the 
former  presentation  or  record  recent  advances.  This  plan  of 
publication  has  been  arranged  in  order  to  meet  the  demand  of 
teachers  and  the  convenience  of  classes,  but  it  is  also  thought 
that  it  may  prove  advantageous  to  readers  in  special  lines  of 
mathematical  literature. 

It  is  the  intention  of  the  pubhshers  and  editors  to  add  other 
monographs  to  the  series  from  time  to  time,  if  the  call  for  the 
same  seems  to  warrant  it.  Among  the  topics  which  are  under 
consideration  are  those  of  elHptic  functions,  the  theory  of  num- 
bers, the  group  theory,  the  calculus  of  variations,  and  non- 
Euclidean  geometry;  possibly  also  monographs  on  branches  of 
astronomy,  mechanics,  and  mathematical  physics  may  be  included. 
It  is  the  hope  of  the  editors  that  this  form  of  pubhcation  may 
tend  to  promote  mathematical  study  and  research  over  a  wider 
field  than  that  which  the  former  volume  has  occupied. 

December,  1905. 


Ill 


AUTHOR'S   PREFACE. 


In  republishing  this  short  treatise  in  book  form  the  author 
solicits  criticism  but  offers  no  apology.  The  type  of  the  book 
he  has  sought  to  imitate  is  that  shown  in  the  "  mathematical 
tracts  "  of  the  late  Sir  George  B.  Air}'.  The  brevity  and  the  con- 
crete illustrations  of  these  "  tracts  "  have  served  very  effectively 
in  introducing  students  to  a  number  of  the  more  difficult  fields 
of  applied  mathematics;  and  it  is  hoped  that  this  treatise  will 
serve  a  similar  end. 

The  theor\'  of  probabihty  and  the  theory  of  errors  now  con- 
stitute a  formidable  body  of  knowledge  of  great  mathematical 
interest  and  of  great  practical  importance.  Though  developed 
largely  through  appHcations  to  the  more  precise  sciences  of  as- 
tronomy, geodesy,  and  physics,  their  range  of  applicabihty  extends 
to  all  of  the  sciences;  and  they  are  plainly  destined  to  play  an 
increasingly  important  role  in  the  development  and  in  the  appli- 
cations of  the  sciences  of  the  future.  Hence  their  study  is  not 
only  a  commendable  element  in  a  liberal  education,  but  some 
knowledge  of  them  is  essential  to  a  correct  understanding  of 
daily  events. 

No  special  novelty  of  presentation  is  claimed  for  this  work; 
but  the  reader  may  find  it  advantageous  to  know  that  a  definite 
plan  has  been  followed.  This  plan  consists  in  presenting  each 
principle,  first,  by  means  of  a  simple,  concrete  example;  passing, 
secondly,  to  a  general  statement  by  means  of  a  formula;  and, 
thirdly,  illustrating  appHcations  of  the  formula  by  concrete 
examples.  Great  pains  have  been  taken  also  to  secure  clear  and 
correct  statements  of  fundamental  facts.  If  these  latter  are 
duly  understood,  the  student  needs  Httle  additional  aid;  if 
they  arc  not  duly  understood,  no  amount  of  aid  will  forward  him. 

The  passage  from  the  elementary  concrete  to  the  advanced 
abstract  may  appear  to  be  abrupt  to  the  reader  in  some  cases. 
It  is  hoped,  however,  that  any  large  gaps  may  be  easily  bridged 
and  that  any  serious  difficulties  may  be  easily  overcome  by  means 
of  the  references  given  to  the  literature  of  the  subject.  In  any 
event  the  student  will  find  that  in  this,  as  in  all  of  the  more  ardu- 
ous sciences,  his  greatest  pleasure  and  his  highest  discipline  will 
come  from  bridging  such  gaps  and  from  surmounting  such  diffi- 
culties. 

Washington,  D.  C,  December,  1905. 


CONTENTS. 


Art.    I.  Introduction Page  7 

2.  Permutations 11 

3.  Combinations 13 

4.  Direct  Probabilities 16 

5.  Probability  of  Concurrent  Events 19 

6.  Bernoulli's  Theorem 22 

7.  Inverse  Probabilities 24 

8.  Probabilities  of  Future  Events 27 

9.  Theory  of  Errors 30 

ic.  Laws  of  Error 3^ 

11.  Typical  Errors  of  A  System 33 

12.  Laws  of  Resultant  Error 34 

13.  Errors  of  Interpolated  Values 37 

14.  Statistical  Test  of  Theory 44 


PROBABILITY  AND  THEORY  OF  ERRORS. 


Art.  1.    Introduction, 

It  is  a  curious  circumstance  that  a  science  so  profoundly 
mathematical  as  the  theory  of  probability  should  have  origi- 
nated in  the  games  of  chance  which  occupy  the  thoughtless  and 
the  profligate.*  That  such  is  the  case  is  sufficiently  attested 
by  the  fact  that  much  of  the  terminology  of  the  science  and 
many  of  its  familiar  illustrations  are  drawn  directly  from  the 
vocabulary  and  the  paraphernalia  of  the  gambler  and  the  trick- 
ster. It  is  somewhat  surprising,  also,  considering  the  antiquity 
of  games  of  chance,  that  formal  reasoning  on  the  simpler 
questions  in  probability  did  not  begin  before  the  time  of  Pascal 
and  Fermat.  Pascal  was  led  to  consider  the  subject  during  the 
year  1654  through  a  problem  proposed  to  him  by  the  Chevalier 
de  Mere,  a  reputed  gamblcr.f  The  problem  in  question  is 
known  as  the  problem  of  i)oints  and  may  be  stated  as  follows: 
two  pla3^ers  need  each  a  given  number  of  points  to  win  at  a 
certain  stage  of  their  game;  if  they  stop  at  this  stage,  how  should 
the  stakes  be  divided  ?  Pascal  corresponded  with  his  friend 
Fermat  on  this  question  ;  and  it  appears  that  the  letters  which 
passed  between  them  contained  the  earliest  distinct  formulation 
of  principles  falling  within   the  theory  of  probability.     These 

*  The  historical  facts  referred  to  in  this  article  are  drawn  mostly  from  Tod- 
hunter's  History  of  the  Mathematical  Theory  of  Probability  from  the  time  of 
Pascal  to  thai  of  Laplace  (Cambridge  and  London,  1865). 

f  "  Un  problcme  relatif  aux  jeux  de  hasard,  propose  a  un  austere  janseniste 
par  un  homme  du  monde,  a  ete  I'origine  du  calcul  des  probabilites. "  Poisson, 
Recherches  surla  Probabilite  des  Jugements  (Paris,  1837). 


8  PROBABILITY  AND  THEORY  OF  ERRORS. 

acute  thinkers,  however,  accompHshed  little  more  than  a  correct 
start  in  the  science.  Each  seemed  to  rest  content  at  the  time 
with  the  approbation  of  the  other.  Pascal  soon  renounced 
such  mundane  studies  altogether  ;  Fermat  had  only  the  scant 
leisure  of  a  life  busy  with  affairs  to  devote  to  mathematics; 
and  both  died  soon  after  the  epoch  in  question, — Pascal  in 
1662,  and  Fermat  in  1665. 

A  subject  which  had  attracted  the  attention  of  such  dis- 
tinguished mathematicians  could  not  fail  to  excite  the  interest 
of  their  contemporaries  and  successors.  Amongst  the  former 
Huygens  is  the  most  noted.  He  has  the  honor  of  publishing 
the  first  treatise*  on  the  subject.  It  contains  only  fourteen 
propositions  and  is  devoted  entirely  to  games  of  chance,  but  it 
gave  the  best  account  of  the  theory  down  to  the  beginning  of 
the  eighteenth  century,  when  it  was  superseded  by  the  more  elab- 
orate works  of  James  Bernoulli, f  Montmort,:};  and  De  Moivre.§ 
Through  the  labors  of  the  latter  authors  the  mathematical 
theory  of  probability  was  greatly  extended.  They  attacked, 
quite  successfully  in  the  main,  the  most  difficult  problems; 
and  great  credit  is  due  them  for  the  energy  and  ability  dis- 
played in  developing  a  science  which  seemed  at  the  time  to 
have  no  higher  aim  than  intellectual  diversion. ||  Their  names, 
undoubtedly,  with  one  exception,  that  of  Laplace,  are  the  most 
important  in  the  history  of  probability. 

Since  the  beginning  of  the  eighteenth  century  almost  every 
mathematician  of  note  has  been  a  contributor  to  or  an  expos- 
itor of  the  theory  of  probability.  Nicolas,  Daniel,  and  John 
Bernoulli,  Simpson,  Euler,  d'Alembert,  Bayes,  Lagrange,  Lam- 
bert, Condorcet,  and  Laplace  are  the  principal  names  which 
figure  in  the  history  of  the  subject  during  the  hundred  years 

*  De  Ratiociniis  in  Ludo  Aleae,  1657. 

t  Ars  Conjectandi,  1713. 

lEssai  d'Analyse  sur  les  Jeux  de  Hazards,  1708. 

§The  Doctrine  of  Chances,  1718. 

ITodhunter  says  of  Montmort,  for  example,  "In  1708  he  published  his 
worlc  on  Chances,  where  with  the  courage  of  Columbus  he  revealed  anew  world 
to  Mathematicians." 


INTRODUCTION.  9 

ending  with  the  first  quarter  of  the  nineteenth  century.  Of 
contributions  from  this  brilliant  array  of  mathematical  talent, 
the  Theorie  Analytique  des  Probabilites  of  Laplace  is  by  far 
the  most  profound  and  comprehensive.  It  is,  like  his  M^- 
canique  Celeste  in  dynamical  astronomy,  still  the  most  elabo- 
rate treatise  on  the  subject.  An  idea  of  the  grand  scale  of  the 
work  in  its  present  form*  maybe  gained  by  the  facts  that  the 
non-mathematical  introductionf  covers  about  one  hundred  and 
fifty  quarto  pages;  and  that,  in  spite  of  the  extraordinary 
brevity  of  mathematical  language,  the  pure  theory  and  its  ac- 
cessories and  applications  require  about  six  hundred  and  fifty 
pages. 

From  the  epoch  of  Laplace  down  to  the  present  time  the 
extensions  of  the  science  have  been  most  noteworthy  in  the 
fields  of  practical  applications,  as  in  the  adjustment  of  obser- 
vations, and  in  problems  of  insurance,  statistics,  etc.  Amongst 
the  most  important  of  the  pioneers  in  these  fields  should 
be  mentioned  Poisson,  Gauss,  Bessel,  and  De  Morgan.  Nu- 
merous authors,  also,  have  done  much  to  simplify  one  or  an- 
other branch  of  the  subject  and  thus  bring  it  within  the  range 
of  elementary  presentation.  The  fundamental  principles  of 
the  theory  are,  indeed,  now  accessible  in  the  best  text-books 
on  algebra  ;  and  there  are  many  excellent  treatises  on  the  pure 
theory  and  its  various  applications. 

Of  all  the  applications  of  the  doctrine  of  probability  none 
is  of  greater  utility  than  tlie  theory  of  errors.  In  astronomy, 
geodesy,  physics,  and  chemistry,  as  in  every  science  which  at- 
tains precision  in  measuring,  weighing,  and  computing,  a 
knowledge  of  the  theory  of  errors  is  indispensable.  By  the  aid 
of  this  theory  the  exact  sciences  have  made  great  progress  dur- 

*The  form  of  the  third  edition  published  in  1S20,  and  of  Vol.  VII  of  the 
complete  works  of  Laplace  recently  republished  under  the  auspices  of  the 
Acad6mie  des  Sciences  by  Gauthier-Villars.     This  Vol.  VII  bears  the  date  1886. 

f  "  Cette  Introduction,"  writes  Laplace,  "  est  le  developpement  d'une  Le5on 
sur  les  Probabilites,  que  je  donnai  en  1795,  aux  Ecoles  Normales,  oil  je  fus  ap- 
pele  comme  professeur  de  Mathematiques  avec  Lagrange,  par  un  decret  de  la 
Convention  nationale." 


10  PROBABILITY    AND    THEORY    OF    ERRORS. 

ing  the  nineteenth  century,  not  only  in  the  actual  determination 
of  the  constants  of  nature,  but  also  in  the  fixation  of  clear 
ideas  as  to  the  possibilities  of  future  conquests  in  the  same  di- 
rection. Nothing,  for  example,  is  more  satisfactory  and  in- 
structive in  the  history  of  science  than  the  success  with  which 
the  unique  method  of  least  squares  has  been  applied  to  the 
problems  presented  by  the  earth  and  the  other  members  of  the 
solar  system.  So  great,  in  fact,  are  the  practical  value  and 
theoretical  importance  of  the  method  of  least  squares,  that  it  is 
frequently  mistaken  for  the  whole  theory  of  errors,  and  is 
sometimes  regarded  as  embodying  the  major  part  of  the  doc- 
trine of  [probability  itself. 

As  may  be  inferred  from  this  brief  sketch,  the  theory  of 
probability  and  its  more  important  applications  now  constitute 
an  extensive  body  of  mathematical  principles  and  precepts. 
Obviously,  therefore,  it  will  be  impossible  within  the  limits  of 
a  single  condensed  monograph  to  do  more  than  give  an  out- 
line of  the  salient  features  of  the  subject.  It  is  hoped,  how- 
ever, in  accordance  with  the  general  plan  of  the  volume,  that 
such  outline  will  prove  suggestive  and  helpful  to  those  who 
may  come  to  the  science  for  the  first  time,  and  also  to  those 
who,  while  somewhat  familiar  with  the  difficulties  to  be  over- 
tome,  have  not  acquired  a  working  knowledge  of  the  subject. 
Effort  has  been  made  especially  to  clear  up  the  difficulties  of 
the  theory  of  errors  by  presenting  a  somewhat  broader  view  of 
the  elements  of  the  subject  than  is  found  in  the  standard 
treatises,  which  confine  attention  almost  exclusively  to  the 
method  of  least  squares.  This  chapter  stops  short  of  that 
method,  and  seeks  to  supply  those  phases  of  the  theory  which 
are  either  notably  lacking  or  notably  erroneous  in  works 
hitherto  published.  It  is  believed,  also,  that  the  elements  here 
presented  are  essential  to  an  adequate  understanding  of  the 
well-worked  domain  of  least  squares.* 

*The  author  has  given  a  brief  but  comprehensive  statement  of  the  method 
of  least  squares  in  the  volume  of  Geographical  Tables  published  by  the  Sniith- 
sonion  Institution,  1894. 


PERMUTATIONS. 


n 


Art.  2.    Permutations. 

The  formulas  and  results  of  the  theory  of  permutations 
and  combinations  are  often  needed  for  the  statement  and  so- 
lution of  problems  in  probabihties.  This  theory  is  now  to  be 
found  in  most  works  on  algebra,  and  it  w  ill  therefore  suffice 
here  to  state  the  principal  formulas  and  illustrate  their  mean- 
ing by  a  few  numerical  examples. 

The  number  of  permutations  of  ii  things  taken  r  in  a  group 
is  expressed  by  the  formula 

(;/),  =  ;/(;/  -  i)(;/  -  2)  .  .  .  {ii  -  r  -f  i).  (i) 

Thus,  to  illustrate,  the  number  of  ways  the  four  letters  a,  b, 

c,  d  can  be  arranged  in  groups  of  two  is  4 .  3  =  1 2,  and  the  groups 
are 

ab,     ba,     ac,     ca,     ad,     da,     be,     cb,     bd,     db,     cd,     dc. 

Similarly,  the  formula  gives  for 
w  =     3  and  r  =  2,  (3),  =  3-2  =6, 

;/==;"     ;-=  3,         (7)3  =  7.6.5  =  210, 

n  —  \o    "     r  =  6,       (10),  =  10.9.8.  7.6.  5  =  15 1200. 

The  results  which  follow  from  equation  (i)  when  n  and  r 
do  not  exceed  10  each  are  embodied  in  the  following  table  : 

VAi.uiis  OF  Permutations. 


10 

9 

8 

7 

6 

5 

4 

4 

3 

3 

2 

2 

I 

I 

I 

10 

9 

8 

7 

6 

5 

2 

90 

72 

5fi 

42 

30 

20 

12 

6 

2 

3 

720 

504 

33^' 

210 

120 

60 

24 

6 

4 

5040 

3024 

16S0 

840 

360 

120 

24 

5 

30240 

15  I  20 

6720 

2S20 

72iJ 

120 

6 

1 5  I  200 

60480 

20160 

5040 

720 

7 

604800 

1S1440 

40320 

5040 

8 

I S 14400 

362S&0 

40320 

9 

362SS00 

362S80 

10 

362S800 

15 

4 

I 

Sp 

9864100 

986409 

109600 

13699 

1956 

325 

64 

The  use  of  this  table  is  obvious.  Thus,  the  number  of  per- 
mutations of  eight  things  in  groups  of  five  each  is  found  in  the 
fifth  line  of  the  column  headed  with  the  number  8.     It  will  be 


12  PROBABILITY    AND    THEORY    OF    ERRORS. 

noticed  that  the.  last  two  numbers  in  each  column  (excepting 
that  headed  with  i)  are  the  same.  This  accords  with  the  for- 
mula, whicli  gives  for  the  number  of  permutations  of  n  things 
in  groups  of  ;/  the  same  value  as  for  ;/  things  in  groups  of 
(;/  —  i).  It  will  also  be  remarked  that  the  last  number  in  each 
column  of  the  table  is  the  factorial,  n\,  of  the  number  n  at  the 
head  of  the  column.  For  example,  in  the  column  under  7,  the 
last  number  is  5040  =  1.2.3.4.5.6.7  =  7!. 

The  total  number  of  permutations  of  n  things  taken  singly, 
in  groups  of  two,  three,  etc.,  is  found  by  summing  the  numbers 
given  by  equation  (i)  for  all  values  of  r  from  i  to  n.  Calling 
this  total  or  sum  Sp,  it  will  be  given  by 

Sp  =  '2{7i\.  (2) 

To  illustrate,  suppose  n  =  3,  and,  to  fix  the  ideas,  let  the 
three  things  be  the  three  digits  i,  2,  3.  Then  from  the  above 
table  it  is  seen  that  Sp  =  :^-\-6-\-6=  15  ;  or,  that  the  number 
of  numbers  (all  different)  which  can  be  formed  from  those  dig- 
its is  fifteen.  These  numbers  are  I,  2,  3  ;  12,13,21,23,31,32; 
123,  132,  213,  231,  312,  321. 

The  values  of  Sp  for  ;/  =  1,2,...  10  are  given  under  the 
corresponding  columns  of  the  above  table.  But  when  71  is 
large  the  direct  summation  indicated  by  (2)  is  tedious,  if  not 
impracticable.  Hence  a  more  convenient  formula  is  desirable. 
To  get  this,  observe  that  (i)  may  be  written 

if  r  is  restricted  to  integer  values  between  i  and  (n  —  i),  both 
inclusive.  This  suffices  to  give  all  terms  which  appear  in  the 
right-hand  member  of  (2),  since  the  number  of  permutations 
for  r  =  {n  —  1)  is  the  same  as  for  r  =  n.  Hence  it  appears 
that 

^  '      I  I  .  2    '  In  —    ^\\ 


{n  -  I) 
\     '    I       I  .  2   '  (;/  —  I ) !/ 


COMBINATIONS.  13 

But  as  n  increases,  the  series  by  which  7i\  is  here  multiplied 
approximates  rapidly  towards  the  base  of  natural  logarithms ; 
that  is,  towards 

^  =  271 828 1 8  +,         log  ^  =  0.4342945. 
Hence  for  large  values  of  n 

5;i  =  ;^ !  ^,  approximately.*  (3) 

To  get  an  idea  of  the  degree  of  approximation  of  (3),  sup- 
pose n  =  9.     Then  the   computation  runs  thus  (see  values  in 

the  above  table) : 

log 

9!  =  362880         5-5597630 
e  0.4342945 

91^  =  986410         5.9940575 
Sp  =  986409  by  equation  (2). 

The  error  in  this  case  is  thus  seen  to  be  only  one  unit,  or 
about  one-millionth  of  Sp.-f 

Prob.  r.  Tabulate  a  list  of  the  numbers  of  three  figures  each 
which  can  be  formed  from  the  first  five  digits  i,  .  .  .  5.  How  many 
numbers  can  be  formed  from  the  nine  digits  ? 

Prob.  2.  Is  Sf,  always  an  odd  number  for  n  odd  ?  Observe 
values  of  Sp  in  the  table  above. 

Art.  3.    Combinations. 

In  permutations  attention  is  given  to  the  order  of  arrange- 
ment of  the  things  considered.  In  combinations  no  regard  is 
paid  to  the  order  of  arrangement.  Thus,  the  permutations  of 
the  letters  a,  d,  c,  d  in  groups  of  three  are 

{abc)  (abd)  bac  bad  acb  {acd)  cab  cad 
adb  adc  dab  dac  bca  {bed)  cba  cbd 
bda       bdc       dba     dbc     cda      cdb       dca     deb 

*  See  Art.  6  for  a  formula  for  computing  n\  when  n  is  a  large  number. 

f  When  large  numbers  are  to  be  dealt  with,  equations  (i)'  and  (3)  are  easily 
managed  by  logarithms,  especially  if  a  table  of  values  of  log  (;/!)  is  available. 
Such  tables  are  given  to  six  places  in  De  Morgan's  treatise  on  Probability  in 
the  Encyclopaedia  Metropolitana,  and  to  five  places  in  Shortrede's  Tables 
(Vol.  I,  1849). 


14  PROnABlLITV  AND  THEORY  OF  ERRORS. 

Rut  if  the  order  of  arrangement  is  ignored  all  of  these  are 
seen  to  be  repetitions  of  the  groups  enclosed  in  parentheses, 
namely,  {abc),  {abd),  [acd),  [bed).  Hence  in  this  case  out  of 
twent}'-four  permutations  there  are  only  four  combinations. 

A  general  fcjrmula  for  computing  the  number  of  combina- 
tions of  ;/  things  taken  in  groups  of  r  things  is  easily  derived. 
For  the  number  of  permutations  of  n  things  in  groups  of  r  is 
b\-  (i)  of  Art.  2 

{,!),.  =  ;/(;/  -   l)(;/  _  2)  ...(;/-  r  +  i) ; 

and  since  each  group  of  r  things  gives  i  .  2  .  3  .  .  .  r  =  r!  per- 
mutations, the  number  of  combinations  must  be  the  cjuotient 
of  (//),.  by  r\.  Denote  this  number  by  C[u)^.  Then  the  gen- 
eral formula  is 

This  formula  gives,  for  example,  in  the  case  of  the  four  let- 
ters a,  b,  c,  d  taken   in  groups  of  three,  as  considered  above, 

4.3.2 

Multi[)ly  both  numerator  and  denominator  of  the  right-hand 
member  of  (i)  b}-  {n  —  r) !     The  result  is 

which  shows  that  the  number  of  combinations  of  n  things  in 
groups  of  r  is  the  same  as  the  number  of  combinations  of  n 
things  in  groups  of  (;;  —  r).  Thus,  the  number  of  combina- 
tions of  the  first  ten  letters  a,  b,  c  .  .  .j  in  groups  of  three  or 

seven  is 

10! 

-— — -  —  120. 

3'- 7! 
The  following  table  gives  the  values  C{ii)^  for  all  values  of 
;/  and  r  froin  I  to  10. 

The  mode  of  using  this  table  is  evident.  For  example,  the 
number  (^f  combinations  of  eight  things  in  sets  of  five  each  is 
found  on  the  fifth  line  of  the  column  headed  8  to  be  56. 


combinations. 
Values  of  Combinations. 


15 


10 

9 

8 

7 

6 

5 

4 

3 

2 

I 

I 

lO 

9 

8 

7 

6 

5 

4 

3 

2 

I 

2 

45 

36 

28 

21 

15 

10 

() 

3 

I 

3 

I20 

84 

56 

35 

20 

10 

4 

I 

4 

2IO 

126 

70 

35 

15 

5 

I 

252 

126 

56 

21 

b 

I 

6 

210 

84 

28 

7 

I 

7 

120 

36 

8 

I 

8 

45 

9 

I 

9 

10 

I 

10 

I 

Sc 

1023 

511 

255 

127 

03 

31 

15 

7 

3 

It  will  be  observed  that  the  numbers  in  any  column  show 
a  maximum  value  when  n  is  even  and  two  equal  maximum 
values  when  //  is  odd.  That  this  should  be  so  is  easily  seen 
from  (i)',  which  shows  that  C{/i),.  will  be  a  maximum  for  any 
value  of  ;/  when  ;■ !  (//  —  ;')  !  is  a  minimum.  For  ;/  even  this  is 
a  minimum  for  r  =  jyJi ;  while  for  n  odd  it  has  equal  minimum 
values  for  r  =  l{/i  —  i)  and  r  —  \{ii  -\-  i).     Thus, 


maximum  of  C{ii)r 


n 


n 


7,  for  ;/  even, 


n\ 


(2) 


n-]-  I  ,  Ji  -  1  , 


for  ;/  odd. 


2  2 

The  total  number  of  combinations  of  ?/  things  taken  singly, 
in  groups  of  two,  three,  etc.,  is  found  by  summing  the  numbers 
given  by  (i)  for  all  values  of  r  from  I  to  ;/  both  inclusive. 
Calling  this  total  or  sum  S^, 

The  same  sum  will  also  come  from  (i)'  by  giving  to  rail  values 
from  I  to  {?i  —  i),  both  inclusive,  summing  the  results,  and  in- 
creasing their  aggregate  by  unity.     Thus  by  either  process 


I  .  2 


1.2.3 


IG  PROBABILITY  AND  THEORY  OF  ERRORS. 

The   second  member    of    this  equation   is  evidently  equal    to 

(i  +  0"  ~~  I-     Hence 

S,  =  2C{fi),.  =  2"  -  I.  (3) 

The  values  of  S,  for  values  of  ;/  and  r  from  i  to  lo  are  given 
under  the  corresponding  columns  of  the  above  table. 

Prob.  3.  How  many  different  squads  of  ten  men  each  can  be 
formed  from  a  company  of  100  men  ? 

Prob.  4  How  many  triangles  are  formed  by  six  straight  lines 
each  of  which  intersects  the  other  five  ? 

Prob.  5.  Examine  this  statement  :  "  In  dealing  a  pack  of  cards 
the  number  of  hands,  of  thirteen  cards  each,  which  can  be  produced 
is  635  013  559  600.  But  in  whist  four  hands  are  simultaneously  held, 
and  the  number  of  distinct  deals  .  .  .  would  require  twenty-eight 
figures  to  express  it."  * 

Prob.  6.  Assuming  combination  always  possible,  and  disregarding 
the  question  of  proportions,  find  how  many  different  substances 
could  be  produced  by  combining  the  seventy-three  chemical  ele- 
ments. 

Art.  4.     Direct  Probabilities. 

If  it  is  known  that  one  of  two  events  must  occur  in  any 
trial  or  instance,  and  that  the  first  can  occur  in  a  ways  and  the 
second  in  d  ways,  all  of  which  ways  are  equally  likely  to  hap- 
pen, then  the  probability  that  the  first  will  happen  is  expressed 
mathematically  by  the  fraction  a/{a-\~/^),  while  the  probability 
that  the  second  will  happen  is  /?/{a  -\-  b).  Such  events  are  said 
to  be  mutually  exclusive.  Denote  their  probabilities  hy  p  and 
q  respectively.     Then  there  result 

the  last  equation  following  from  the  first  two  and  being  the 
mathematical  expression  for  the  certainty  that  one  of  the  two 
events  must  happen. 

Thus,  to  illustrate,  in  tossing  a  coin  it  tnust  give  "  head  "  or 
"  tail"  ;  ^  =  /;  =  I,  and  /^  =  r/  =  1/2.  Again,  if  an  urn  contain 
^  =:  5  white  and  /5  =  8  black  balls,  the  probability  of  drawing 

*  Jevons,   Principles  of  Science,  New  York,  1S74,  p.  217. 


blRECT    rROlSABlLITlES.  17 

a  white  ball  in  one  trial  is  p  =  5/13  and  that  of  drawing  a 
black  one  q  =  8/13. 

Similarly,  if  there  are  several  mutually  exclusive  events 
which  can  occur  in  a,  b,  c .  .  .  wa}-s  respectively,  their  probabil- 
ities/, (],  r  .  .  .    are  given  by 

a  b  c 

^  "^  ~a-\-b-^c--Y.  .' '  ^  ^  a-^b^c-\-.  .  . '  ^  ^  a^b^c-\-... ' 

(2) 
/  +  (7  +  ^  +  -  •  •=  I- 

For  example,  if  an  urn  contain  ^7=4  white,  <^  =  5  black, 
and  (f  ^  6  red  balls,  the  probabilities  of  drawing  a  white,  black, 
and  red  ball  at  a  single  trial  are  ^  =  4/15,  q  =.  5/15,  and 
r  —  6/15,  respectively. 

Formulas  (i)  and  (2)  may  be  applied  to  a  wide  variety  of 
cases,  but  it  must  suffice  here  to  give  only  a  few  such.  As  a 
first  illustration,  consider  the  probability  of  drawing  at  random 
a  number  of  three  figures  from  the  entire  list  of  numbers  which 
can  be  formed  from  the  first  seven  digits.  A  glance  at  the 
table  of  Art.  i  shows  that  the  s\'mbols  of  formula  (1)  have  in 
this  case  the  values  a  =■  210,  and  <'r -j- /;  =  13699.  Hence 
b  =  13489,  and  /  =  210/13699  ;  that  is,  the  probability  in  ques- 
tion is  about  1/65. 

Secondly,  what   is  the  probability  of  holding  in  a  hand   of 

whist  all  the  cards  of  one  suit  ?     Formula  (i)  of  Art.  3  shows 

that  the  number  of  different  hands  of  thirteen  cards  each  which 

may  be  formed  from  a  pack  of  fifty-two  cards  is 

52. 5 1. 50... 40       ^  ^ 

- — - — =^ ^—  =  635  o 1 3  5 59  600, 

I  . 2  .  3  ...  13  ^^      ^  ^^^ 

and  the  probability  required  is  the  reciprocal  of  this  number. 
The  probability  against  this  event  is,  therefore,  very  nearly 
unit)'. 

Thirdly,  consider  the  probabilities  presented  by  the  case  of 
an  urn  containing  4  white,  5  black,  and  6  red  balls,  from  which 
at  a  single  trial  three  balls  are  to  be  drawn.  E\'idcntl\'  the 
triad  of  balls  drawn  may  be  all  white,  all  black,  all  red,  partly 
white  and  black,  partly  white  and  red,  partly  black  and  red,  or 


18  PROIlAIilLlTV    AND    THEORY    OF    ERRORS. 

one  each  of  the  white,  bLick,  and  red.  There  are  thus  seven 
different  probabihties  to  be  taken  into  account.  The  theory 
of  combinations  shows  (see  equation  (i),  Art.  3)  that  the  total 

number  of 

4.^.2 

White  triads  =    =     4  =  a 

1.2.3 

C        A        ^ 

Black  triads  =         .; —  =   10  =  (5 

6 

6.5.4 
Red  triads  =    — 7 =   20  —  c 


I  . 

2  . 

3 

5. 

.4. 
6 

^ 

6, 

•5. 

4 

6 

9 

.8. 

7 

b 

10 

■9 

.8 

6 

1 1 

.10.9 

White  and  black  triads  =:    — —  (  4+10)=  70  =  d 

10.  9 . 8       ^ 
White  and  red  triads  =  ^ —  (  4-I-20)—  96  =  ^ 

Black  and  red  triads  1^  — '—, — '- (io-[-2o)=  135  =y 

White,  black,  and  red  triads  ^4.5.6  :=I20—  ^ 

Sum  =  455 
The  total  number  of  these  triads  is  455,  and  is,  as  it  should 
be,  the  number  of  combinations  in  groups  of  three  each  of  the 
whole  number  of  balls.  Hence  formulas  (2)  give  the  seven 
different  probabilities  which  follow,  using  the  initial  letters 
w,  d,  r  to  indicate  the  colors  represented  in  a  triad  : 
For  a  triad  ivtmv  p  =      4/455, 

"    "     "       bbb  q  =     10/455, 

"     "      "       rrr  r  =     20/455, 

"    "     "       7V'ci'b  or  7c>bb  s  =    70/455, 

"     "     "       7V7vr  or  tvrr  t  =    96/455, 

"     "     "       bbr  ov  brr  //  =  135/455, 

"     "     "       wbr  V  =  120/455. 

Prol).  7.  Wlien  three  dice  are  thrown  together,  what  is  the  prob- 
ability that  tlie  tlirow  will  be  greater  than  9  ? 

Prol).  8.  Write  down  a  literal  fornuila  for  the  probabilities  of  the 
several  ])ossible  triads  considered  in  the  above  question  of  the  balls, 
sui)i)osing  the  numbers  of  white,  black,  and  red  balls  to  be  /,  ;«.  n, 
respectively. 


PROBABILITY    OF    CONCURRENT    EVENTS.  19 

Art.  5.     Probability  of  Concurrent  Events. 

If  the  probeibilities  of  two  independent  events  are/,  and 
p.^,  respectively,  the  probability  of  their  concurrence  in  any 
single  instance  is  pp^.  Thus,  suppose  there  are  two  urns 
U^  and  U^,  the  first  of  which  contains  a^  white  and  l\  black 
balls,  and  the  second  a^  white  and  b^  black  balls.  Then  the 
probability  of  drawing  a  white  ball  from  U^  is/,  =  aj{(i^  -}~  '^i)> 
while  that  of  drawing  a  white  ball  from  U^  is/^  :=  ^'^Ji^^^i  +  b^). 
The  total  number  of  different  pairs  of  balls  which  can  be  formed 
from  the  entire  number  of  balls  is  {a^  -\-  h^){a^  -j"  ^^-  ^^  these 
pairs  a^a^  are  favorable  to  the  concurrence  of  white  in  simul- 
taneous or  successive  drawings  from  the  two  urns.  Hence  the 
probability  of  a  concurrence  of 

white  with  white  =  a^a^/{a^  -j-  b^){a^  -\-  b^, 

white  with  black  =  {aj?^'\-a^b^)/{a,  -\-  b,){a^  -\-  b^, 

black  with  black  =  b,b^/{a,  +  <^,)(^,  -f  /7J, 

and  the  sum  of  these  is  unity,  as  required  by  equations  (2)  of 
Article  4. 

In  general,  if /,, /j, /j  .  .  .  denote  the  probabilities  of  several 
independent  events,  and  P  denote  the  probability  of  theii 
concurrence, 

P  =  P,PiP^"'  '  (i) 

To  illustrate  this  formula,  suppose  there    is  required  the 
probability  of  getting  three  aces  with  three  dice  thrown  simul- 
taneously.    In  this  case/,  =  p^  =  p^  =  1/6  and 
P={\/6f  =  \/2\6. 

Similarly,  if  two  dice  are  thrown  simultaneously  the  proba- 
bility that  the  sum  of  the  numbers  shown  will  be  11  is  2/36; 
and  the  probability  that  this  sum  i  i  will  appear  in  two  succes- 
sive throws  of  the  same  pair  of  dice  is  ^/T)6.t^6. 

The  probability  that  the  alternatives  of  a  series  of  events 
will  concur  is  evident  1\-  given  by 

Q  =  q.q.iz . . .  =  (i  -  A)(i  -A)(i  -A)  •  •  •  (2) 

Thus,  in  the  case  of  the  three  dice  mentioned  above,  the 
probability  that  each  will  show  something  other  than  an  ace  is 


I 


20  PROBABILITY    AND    THEORY    OF    ERRORS. 

q^~  q^—  q^  =  5/6,  and  the  probability  that  they  vvill  concur  in 
this  is  (2  =  125/216. 

Many  cases  of  interest  occur  for  the  apphcation  of  (i)  and 
(2).  One  of  the  most  important  of  these  is  furnished  by  suc- 
cessive trials  of  the  same  event.  Consider,  for  example,  what 
may  happen  in  n  trials  of  an  event  for  wliich  the  probability 
is  />  and  against  which  the  probability  is  q.  The  probabilit> 
that  the  event  will  occur  every  time  is/".  The  probability  that 
the  event  will  occur  (//  —  I)  times  in  succession  and  then  fail  is 
/"  "  Vy.  But  if  the  order  of  occurrence  is  disregarded  this  last 
combination  may  arrive  in  ;/  different  ways;  so  that  the  prob- 
abilit)'  that  the  event  will  occur  (;/  —  I)  times  and  fail  once  is 
iip"~\j.  Similarly,  the  probability  that  the  event  will  happei* 
(/^  —  2)  times  and  fail  twice  is  \n{}i  —  •)/  'V  !  ^^^'  That  is, 
the  probabilities  of  the  several  possible  occurrences  are  given  b)'  „ 
the  corresponding  terms  in  the  development  of  (/  +  q)".  " 

By  the  same  reasoning  used  to  get  equations  (2)  of  Art, 
3  it  may  be  shown  that  the  maximum  term  in  the  expansion 
of  (/H-f/)"  is  that  in  which  the  exponent  ;;/,  say,  of  q  is 
the  whole  number  lying  between  {u -\-  \)q  —  i  and  {n -\-  i)q.  ^ 
In  other  words,  the  most  probable  result  in  «  trials  is  the 
occurren'ce  of  the  event  {n  —  m)  times  and  its  failure  7n 
times.  When  n  is  large  this  means  that  the  most  probable  of 
all  possible  results  is  that  in  which  the  event  occurs  u  —  nq 
—  ii{\  —  q)  z=  np  times  and  fails  7/(7  times.  Thus,  if  th.e  event 
be  that  of  throwing  an  ace  with  a  single  die  the  most  probable 
of  the  possible  results  in  600  throws  is  that  of  100  aces  and 
500  failures. 

Since  q''  is  the  probability  that  the  event  will  fail  every  time 
in  //  trials,  the  [)robability  that  it  will  occur  at  least  once  in  n 
trials  is  I  —  q'\     Calling  this  probability  r,* 

r=  I  -^"=  I  -(i  -p)\  (3) 

If  r  in  this  equation  be  replaced  by  1/2,  the  corresponding 
value    of    n  is  the   number    of    trials   essential   to   render  the 

*  See  Poisson's   Probabilite  des  Jugements,  pp.  40,  41. 


I 


PROBABILITY    OF    CONCURRENT    EVEN  IS.  21 

chances  even  that  the  event  whose  probabihty  is/  will  occur 

at  least  once.     Tiiiis,  in  this  case,  the  value  of  n  is  given  by 

log  2 
;/  =  — 


log  (I-/)' 

This  shows,  for  example,  if  the  event  be  the  throwing  of  double 
sixes  with  two  dice,  for  which  /  =  1/36,  that  the  chances  are 
even  (r  =  1/2)  that  in  25  throws  {ji  —  24.614  by  the  formula) 
double  sixes  will  appear  at  least  once. 

Equation  (3)  shows  that  however  small/  may  be,  so  long  as 
it  is  finite,  n  may  be  taken  so  large  as  to  make  r  approach  in- 
definitely near  to  unity  ;  that  is,  11  may  be  so  large  as  to  render 
it  practically  certain  that  the  event  will  occur  at  least  once. 

When  71  is  large 

(.       M«       T        ...    I   '^(^^~  ^)>.^     ;/(;?-  l){n-2) 

(I  -/)"  =1  -  np  -\ :^—^P TTTs ^  +  •  •  • 

=  I  —  np  A h  .  •  . 

■'      '      1.2  1.2.3' 

=  e~"^  approximately. 

Thus  an  approximate  value  of  r  is 

r=  I  —  e-  "f,         \ogc=  0.4342495.  (4) 

This  formula  gives,  for  example,  for  the  probability  of  drawing 
the  ace  of  spades  from  a  pack  of  fifty-two  cards  at  least  once  in 
104  trials  r=  i  —^~'— 0.865,  while  the  exact  formula  (3) 
gives  0.867. 

Similarly,  the  probability  of  the  occurrence  of  the  event  at 
least  /  times  in  ;/  trials  will  be  given  by  the  sum  of  the  terms 
of  (/  4-  (])"  from  p"  up  to  that  in  p'g"~*  inclusive.  This  proba- 
bility must  be  carefully  distinguished  from  the  probability  that 
the  event  will  occur  /  times  only  in  the  n  trials,  the  latter  being 
expressed  by  the  single  term  in  p'q"~^. 

Prob.  g.  Compare  the  probability  of  holding  exactly  four  aces  in 
five  liands  of  whist  with  the  probability  c5  'lolding  at  least  four  aces 
in  the  same  number  of  hands. 

Prob.  10.  What  is  the  probability  of  an  event  if  the  chances  are 
even  that  it  occurs  at  least  once  in  a  million  trials?  See  equation  (4). 


T=  '^'  -  ■'<"  -:^  ■  •  •  ^^'  +  'W  =  r^,r<r.    (.) 


22  PROBABILITY  AND  THEORY  OF  ERRORS. 

Art.  6.    Bernoulli's  Theorem. 

Denote  the  exponents  of/  and  q  in  the  maximum  term  of 
(^pj^gY  by  //  and  iii  respectively,  and   denote  this  term  by  T. 

Then 

I 

As  shown  in  Art.  5,  /<  in  this  formula  is  the  greatest  whole 
number  in  {;/  -}-  i^p,  and  ///  the  greatest  whole  number  in 
(^jiJ^  i)^;  so  that  when  )i  is  large,  /t  and  in  are  sensibly  equal 
to  ;//  and  nq  respectively. 

The  direct  calculation  of  T  by  (i)  is  impracticable  when  n 
is  large.  To  overcome  this  difficulty  the  following  expression 
is  used  :* 

.!  =  ;/V-"v'2^(i+:^  +  ^ +...).  (2) 

log  ^=0.4342495,       log  2.T=  0.7981799. 


This  expression  approaches  n"c"'  Vmn  as  a  limit  with  the 
increase  of  //,  and  in  this  approximate  form  is  known  as  Stir- 
ling's theorem.  Although  a  rude  approximation  to  // !  for 
small  values  of  //  this  theorem  suffices  in  nearly  all  cases 
wherein  such  probabilities  as  7"  are  desired.  Making  use  of 
the  theorem  in  (i)  it  becomes 

V  27tiipq 

That  this  formula  affords  a  fair  approximation  even  when 
n  is  small  is  seen  from  the  case  of  a  die  thrown  12  times.  The 
probability  that  any  particular  face  will  appear  in  one  thrt)w  is 
p  =:  1/6,  whence  q  =  5/6;  and  the  most  probable  result  in  12 
throws  is  that  in  which  the  particular  face  appears  twice  and 
fails  to  appear  ten  times.  The  probability  of  this  result  com- 
puted from  (3)  is  0.309,  while  the  exact  formula  (i)  gives  0.296. 

The  probability  that  the  event  will  occur  a  number  of  times 

*  This  expression  is  due  to  Laplace,  Tli6oiie  .•\nalytique  des  Probabilit^s. 
See  also  De  Morgan's  Calci^lus,  pp.  600-604. 


Bernoulli's  theorkm.  33 

comprised  between  (/<  —  a)  and  (//  -[-  c^)  in  /i  trials  is  evidently 
expressed  by  the  sum  of  the  terms  in  {p  +  g)"  for  which  the 
exponent  of />  has  the  specified  range  of  values.  Calling  this 
probabilit)'  A',  jniuing 

1^1  =1  lip  -j-  //,     and      in  =  iiq  —  //, 

and  using  Stirling's  theorem  (which  implies  that  //  is  a  large 
number),^ 


R  =  2-=-\i  J^  -r]  I 

V2n//pcj^         npl  \         nql 


> 


very  nearly ;  and  the  summation  is  with  respect  to  ti  from 
//  =  —  a  \.o  i(  ^^  ^  a.  But  expansion  shows  that  the  natural 
logarithm  of  the  product  of  the  two  binomial  factors  in  this 
equation  is  approximately  —  if /2npq.     Hence 

R  =  :2 — ^ — ^-"'^/a"/? . 

\'2n)ipq 

and.  since  n  is  supposed  large,  this  may  be  replaced  by  a  definite 
integral,  putting 


Thus 


dz  —  i/Vinpq,     and     s"  =  n"  /2npq. 

-y  a/  ^%npq  a/  ^Inpq 


R 


=  -^   /  c-'-\{z  =  -^    /  i—dz.  (4) 


Tliis  equation  expresses  the  theorem  of  James  Bernoulli, 
given  in  iiis  Ars  Conjectandi,  published  in   1713. 

The  value  of  the  right-h.and  member  of  (4)  varies,  as  it 
should,  between  o  and  I,  and  ap[)roaches  the  latter  limit  rap- 
idly as  s  increases.     Thus,  writing  for  brevity 


hi-. 


V? 


0 


*  See  Bertrand,  Calcul  des  Probabilites,  Paris,  1889,  for  an  extended  discus- 
sion of  the  questions  considered  in  this  Article. 


24 


PROBABILITY    AND    THEORY    OF    ERRORS. 


the  followiiigr  table  shows  the  march  of  the  intec;ral 


z 

/ 

z 

I 

z 

I 

o.oo 

0.000 

0.75 

0.7II 

1.50 

0.966 

•25 

.276 

1. 00 

.843 

I  75 

.987 

•50 

.520 

1-25 

•923 

2.00 

•995 

To  iUustrate  the  use  of  (4),  suppose  there  is  required  the 
probabiUty  that  in  6000  throws  of  a  die  the  ace  will  appear  a 
number  of  times  which  shall  be  greater  than  1/6  X  6000  —  10 
and  less  than  1/6  X  6000--]-  10,  or  a  number  of  times  lying 
between  990  and  loio.  In  this  case  a  =  10,  ;/  =  6000,/  ^  1/6, 
q  —  5/6.  Thus,  a/V2iipq  =  10/V2  .  6000  .  1/6.  5/6  =  0.245. 
Hence,  by  (4)  and  the  table,  R  —  0.27. 

Prob.  II.  If  the  ratio  of  males  to  females  at  birth  is  105  to  100, 
what  is  the  probability  that  in  the  next  10,000  births  the  number  of 
males  will  fall  within  two  per  cent  of  the  most  probable  number? 

Prob.  12.  If  the  chance  is  even  for  head  and  tail  in  tossing  a 
coin,  what  is  the  probability  that  in  a  million  throws  the  difference 
between  heads  and  tails  will  exceed  1500? 

Art.  7.     Inverse  Probabilities.* 

If  an  observed  event  can  be  attributed  to  any  one  of  several 
causes,  what  is  the  probability  that  any  particular  one  of  these 
causes  produced  the  event  ?  To  put  the  question  in  a  concrete 
form,  suppose  a  white  ball  has  been  drawn  from  one  of  two 
urns,  f/,  containing  3  white  and  5  black  balls,  and  U^  contain- 
ing 2  white  and  4  black  balls  ;  and  that  the  probability  in  favor 
of  each  urn  is  required.  If  f/,  is  as  likely  to  have  been 
chosen  as  U^,  the  probability  that  U^  was  chosen  is  1/2.  After 
such  choice  the  probability  of  drawing  a  white  ball  from  U^  is 
3/8.  Before  drawing,  therefore,  the  probability  of  getting  a 
white  ball  from  V\  was  1/2  X  3/8  =  3/16,  by  Art.  5.  Similarly, 
before  drawing  the  probability  of  getting  a  white  ball  from  f/ 
was  1/2  X  2/6  =  1/6.  These  probabilities  will  remain  un- 
changed if  the  number  of  balls  in  either  urn  be  increased  or 


I 
I 


See  Poisson,  Probabilite  des  Jugements,  pp.  81-83. 


INVERSE    PROBABILITIES.  25 

diininislied  so  long  as  the  ratio  of  white  to  bhick  balls  is  kept 
constant.  Make  these  numbers  the  same  for  the  two  urns. 
Thus  let  the  first  contain  9  white  and  15  black,  and  the  second 
8  white  and  16  black;  whence  the  above  probabilities  may  be 
written  1/2  X  9/24  and  1/2  X  8/24.  It  is  now  seen  tliat  there 
are  (9  -|-  8)  cases  favorable  to  the  production  of  a  white  ball, 
each  of  which  has  the  same  antecedent  probability,  namely,  1/2. 
Since  the  fact  that  a  white  ball  was  drawn  excludes  considera- 
tion of  the  black  balls,  the  probability  that  the  white  ball  came 
from  U^  is  9/17  and  that  it  came  from  U^  is  8/17;  and  the  sum 
of  these  is  unity,  as  it  should  be. 

To  generalize  this  result,  let  there  be  ni  causes,  (7j,  C,,  .  .  .  C,,,. 
Denote  their  direct  probabilities  by^/,,^,, . , .  q,„\  their  antecedent 
probabilities  b)'  r,,  r,,  .  .  .  r„, ;  and  their  resultant  probabilities 
on  the  supposition  of  separate  existence  by  p^,  p^,  ■  .  .  pm- 
That  is, 

Let  D  be  the  common  denominator  of  the  right-hand  mem- 
bers in  (i),  and  denote  the  corresponding  numerators  of  the 
several  fractions  hy  s.,  s^,  .  .  .  s,„.     Then 

p,  =  sJD,    p^  —  sJD,  .  .  .  /„  =  sJD ; 

and  it  is  seen  that  there  are  in  all  (.?,  -]-  i-,  -|-  •  •  •  ■^»/)  equally 
possible  cases,  and  that  of  these  s^  are  favorable  to  C,,  s^  to 
C^,  .  .  .  Hence,  if  /*,,  P^,  .  .  .  P,^  denote  the  probabilities  of 
the  several  causes  on  the  supposition  of  their  coexistence, 

Thus  in  general 

P,  =  pj^p,       P,  =  pJZip,  .  .  .  P,„  =  pj^p.  (2) 

To  illustrate  the  meaning  of  these  formulas  by  the  above 
concrete  case  of  the  urns  it  suffices  to  observe  that 

for  U„      q,  =  3/8     and     ;-,  =  1/2, 

for  U^,      q^  —  1/3     and     r^  —  1/2  ; 
whence  /,  =  3/16,    /,  =  1/6.     /,  +/,  =  17/48  ; 

and  P^  —  9/17,    P^  =  8/17. 

As  a  second   illustration,  suppose  it  is  known  that  a  white 


26  PROBABILITY  AND  THEORY  OF  ERRORS. 


ball  has  been  drawn  from  an  urn  wliich  originally  contained  ;;/ 
balls,  some  of  them  being  black,  if  all  are  not  white.  What  is 
the  probabilil)'  that  the  urn  contained  exactly  )i  white  balls? 
The  facts  are  consistent  with  m  different  and  eaually  probable 
h\-potheses  (or  causes),  namel\-,  that  there  were  i  white  and 
{jn  —  i)  black  balls,  2  white  and  (;//  —  2)  black  balls,  etc. 
Hence  in  (i),  ^7,  =  ^.^  =  .  .  .  =  i,  and 

/,  =  i/m,    p.,  =  2/7//,  .../'„  =  ///w.  .  .  ./,„  =  in/m. 
Thus  ^/=  (i/2)(;//4-  I), 

"2,11 

and  P^^=zpJ2p=^ ■ -. 

This  shows,  as  it  evidently  should,  that  n  =  in  is  the  most 
probable  number  of  white  balls  in  the  urn.  The  probability 
for  this  number  is  P,,^  —  2/{i/i  -j-  i),  which  reduces,  as  it  ought, 
to  I  for  ///!  =  I. 

Formulas  (i)  and  (2)  may  also  be  applied  to  the  problem  of 
estimating  the  probability  of  the  occurrence  of  an  event  from 
the  concurrent  testimony  of  severed  witnesses,  X^,  X^,  .  .  . 
Denote  the  probabilities  that  the  witnesses  tell  the  truth  by 
X,  A\,  .  .  .  Then,  supposing  them  to  testify  independently, 
the  probabilit}'  that  they  will  concur  in  the  truth  concerning 
the  event  is  .i',.i\  .  .  .',  while  the  probability  that  they  will  con- 
cur in  the  onl}'  other  alternativt.-,  falsehood,  is  (i  — -i'j)(i  —-t\)  . .  . 
The  two  alternatives  are  equally  possible.  Hence  by  equations 
(i)  and  (2) 

/,  =  ,r,.n  .  .  .,     /.,  =  (i  -  ,r,)(i  —  ,rj  .  .  ., 


(3) 


P^  being  the  probabilit}'  for  and  /'.,  that  against  the  events 

To  illustrate  (3),  if  the  chances  are  3  to  I  that  A^,  tells  the 
truth  and  5  to  i  that  X.,  tells  the  truth,  .i\  =  3/4,  x^  =  5/6,  and 
/'  =  15/16;  or,  the  chances  are  15  to  i  that  an  event  occurred 
if  they  agree  in  asserting  that  it  did.* 

*  For  some  interesting  applications  of  equations  (3)  see  note  E  of  Appendix 
to  the  Ninth  Bridgewater  Treatise  by  Charles  Babbage  (London,  1838). 


p  — 

.r,,t',  .  .  . 

p 

.  .-U(l  -.r,)(i  -  X,).  .  . 
(i-a-,)(i  --1-,)... 

.  .  H-(l  -  ,r,)(i  -  x)  .  .  . 

PROCABII-ITIES    OF    FUTURE    EVENTS.  27 

It  is  of  theoretical  interest  to  observe  that  if  x,,  a\,  .  .  .  in 
(3)  are  each  greater  than  1/2,  /',  ap[)roaches  unity  as  the 
number  of  witnesses  is  indefinite!}'  increased. 

Prob.  13.  The  groups  of  numbers  of  one  llgure  each,  two  figures 
each,  three  figures  each,  etc.,  which  it  is  jwssihle  to  form  from  the 
nine  digits  i,  2,  ...  9  are  printed  on  cards  and  ])laced  severally  in 
nine  similar  urns.  What  is  the  probability  that  the  number  777  will 
be  drawn  in  a  single  trial  by  a  person  unaware  of  the  contents  of 
the  urns  ? 

Prob.  14.  How  many  witnesses  whose  credibilities  are  each  3/4 
are  essential  to  make  /',  =  o-999  iri  equation  (3)  ? 

Art.  8.     Probabilities  of  Future  Events. 

Equations  (2)   of   Art.  7   may  be  written   in   the  following 

manner:  ^        „  „ 

P       P  P  \ 

.    •    •  ~^^     •  \  1  / 

/.  /.  Pn.  ^P  ^     ^ 

If /i.  A'  •  •  •  A"  ^''^  found  by  observation,  P^,  P^,  .  .  .  P,„  will  ex- 
press the  probabilities  of  the  corresponding  causes  or  their 
efTects.  When,  as  in  the  case  of  most  plu'sical  facts,  the  num- 
ber of  causes  and  events  is  indefinitely  great,  tiie  value  of  any 
p  ox  P  in  (1)  becomes  indefinitely  small,  and  the  value  of  '^p 
must  be  expressed  by  means  of  a  definite  integral.  Let  x  de- 
note the  probability  of  any  particular  cause,  or  of  the  event  to 
which  it  gives  rise.  Then,  supposing  this  and  all  the  other 
causes  mutually  exclusive,  (i  —  x)  will  be  the  probability 
against  the  event.  Now  suppose  it  has  been  observed  that  in 
{j)i  -f-  ii)  cases  the  event  in  question  has  occurred  ///  times  and 
failed  n  times.  The  probability  of  such  a  concurrence  is,  by 
Art.  5,r,i-"'(i  —  .1')",  where  <r  is  a  constant.  Since  a-  is  unknown, 
it  may  be  assumed  to  have  any  value  within  the  limits  o  and  I  ; 
and  all  such  values  are  a   priori  equally  possible.      Put 

y  =  cx"'{l  —  xy. 
Then   evidently  the  probability  that  x  will   fall  within   any  as- 
signed possible  limits  a  and  h  is  expressed  by  the  fraction 


0  1 

Jydxj  J'ydx-^ 


28  PROBABILITY  AND  THEORY  OF  ERRORS. 

SO  that  the  probability  of  any  particular  x  is  given  by 

P  =     "'"C-")''^.  (2) 


/'.r"'(i  —  xYdx 


This  may  be  regarded  as  the  antecedent  probability  of  the 
cause  or  event  in  question. 

What  then  is  the  probability  that  in  the  next  {r  -\-  s)  trials 
the  event  will  occur  r  times  and  fail  s  times,  if  no  regard  is  had 
of  the  order  of  occurrence?  If  x  were  known,  the  answer 
would  be  by  Arts.  2  and  5 

(r  +  5) ! 


■t    r    I 


r!5!  -^"(I-^)'-  (3) 

But  since  x  is  restricted  only  by  the  condition  (2),  the  required 
probability  will  be  found  by  taking  the  product  of  (2)  and  (3) 
and  integrating  throughout  the  range  of  x.  Thus,  calling  the 
required  probability  (9, 


1 

Ar'"+''(i  —  xY^'dx 

1 

j\"'{\  —  xfdx 


r\s\  ^  \t/ 


The  definite  integrals  which  appear  here  are  known  as  Gamma 
functions.  They  are  discussed  in  all  of  the  higher  treatises  on 
the  Integral  Calculus.  Applying  the  rules  derived  in  such 
treatises  there  results  * 

Jf\-s)\{m-\-ry.{n^sy.{m-\-n^iy. 
^  r\s\m\n\  {m  -[-  n -\- r  -\- s  +i)\       '  ^5) 

If  regard  is  had  to  the  order  of  occurrence  of  the  event ; 
that  is,  if  the  probability  required  is  that  of  the  event  happen- 
ing r  times  in  succession  and  then  failing  .y  times  in  succession, 

*  It  is  a  remarkable  fact  that  formula  (?)  is  true  without  restriction  as  to 
values  of  m,  n,  r,  s.  Tlie  formula  may  be  established  by  elementary  considera- 
tions, as  was  done  by  Prevost  and  Lhuilier,  1795.  See  Todhuiiter's  History  oi 
he  Theory  of    Probability,  pp.  453-457. 


PROBABILITIES    OF    FUTURE    EVENTS.  29 

the  factor  {r -\- s)l /rls\  in  (3),  (4),  (5)  must  be  replaced  by 
unity. 

To  illustrate  these  formulas,  suppose  first  that  the  event 
has  happened  j/i  times  and  failed  no  times.  What  is  the  prob- 
ability that  it  will  occur  at  the  next  trial?     In  this  case  (4) 

gives 

I  I 

Q  =    i\"'+\lv/  l'x"'dx  =  {m  4-  \)/{m  -f-  2). 

0  0 

When  in  is  large  this  probability  is  nearly  unity.  Thus,  the 
sun  has  risen  without  failure  a  great  number  of  times  m ;  the 
probability  that  it  will  rise  to-morrow  is 

l\l       .      2\-'  I      .      2 


\     '    7/1'  \     '    inj  in       irr 

which  is  practically  i. 

Secondly,  suppose  an  urn  contains  white  and  black  balls  in 
an  unknown  ratio.  If  in  ten  trials  7  white  and  3  black  balls 
are  drawn,  what  is  the  probability  that  in  the  next  five  trials 
2  white  and  3  black  balls  will  be  drawn?  The  application  of 
(5)  supposes  the  ratio  of  the  white  and  black  balls  in  the  urn 
to  remain  constant.  This  will  follow  if  the  balls  are  replaced 
after  each  drawing,  or  if  the  number  of  balls  in  the  urn  is  sup- 
posed infinite.     The  data  give 

m  =z  7,  ?/  =  3.  r  =  2,  J  =  3. 

;«  -[-  ^  =  9,     7/  +  5  =  6,     r  -f-  i'  =  5 ,     w  -f-  ;/  4"  I  ==  1 1  > 
in  A;-  n  -{-  r  A^  s  -\-  \  =  16. 

Thus  by  (5) 

5!q!6!ii! 
^  ==2T3-!7!iTT6!  =  30/91. 

Suppose  there  are  two  mutually  exclusive  events,  the  first 
of  which  has  happened  ;//  times  and  the  second  ;/  times  in 
m  -{- n  trials.  What  is  the  probability  that  the  chance  of  the 
occurrence  of  tiie  first  exceeds  1/2  ?  The  answer  to  this  ques- 
tion is  given  directly  by  equation  (2)  by  integrating  the  nume- 
rator between  the  specified  limits  of  x.     That  is. 


30  PROBABILITY  AND  THEORY  OF  ERRORS. 

rx"'{i  —  xfdx 

P  -  -^^^ .  (6) 

j'  x'\\  —  xfdx 

0 

Thus,  if  m  =  I  and  ;:  =  o,  P  =  3/4;  or  the  odds  are  three  to 
one  that  the  event  is  more  likcl\^  to  happen  than  not.  Simi- 
larly, if  the  event  has  occurred  ;//  times  in  succession, 

P  =  I  -  (i/2)'"+S 

which  approaches  unity  rapidly  with  increase  of  n. 

Art.  9.     Theory  of  Errors. 

The  theory  of  errors  may  be  defined  as  that  branch  of  math- 
ematics which  is  concerned,  first,  with  the  expression  of  the  re- 
sultant effect  of  one  or  more  sources  of  error  to  which  com- 
puted and  observed  quantities  arc  subject  ;  and,  secondly,  with 
the  determination  of  the  relation  between  the  magnitude  of 
an  error  and  the  probability  of  its  occurrence.  In  the  case  of 
computed  quantities  which  depend  on  numerical  data,  such  as 
tables  of  logarithms,  trigonometric  functions,  etc.,  it  is  usually 
possible  to  ascertain  the  actual  values  of  the  resultant  errors. 
In  the  case  of  observed  quantities,  on  the  other  hand,  it  is  not 
generally  possible  to  evaluate  the  resultant  actu'al  error,  since 
the  actual  errors  of  observation  are  usually  unknown.  In  either 
case,  however,  it  is  always  possible  to  write  down  a  symbolical 
expression  which  A\'ill  show  how  different  sources  of  error  enter 
and  affect  the  aggregate  error;  and  the  statement  of  such  an 
expression  is  of  fundamental  importance  in  llie  theory  of  errors. 

To  fix  the  ideas,  suppose  a  quantit)'  (7  to  be  a  function   of 
several  independent  quantities  x,  y,  s  .  .  .;  that  is, 

Q=f{x,y,,...), 
and  let  it  be  required  to  determine  the  error  in  Q  due  to  errors 
in    X,  y,  ,cr  .  .  .       Denote    such    errors    by  AQ,  Ax,  Ay,  Ac  .  .  . 
Then,  supposing  the  errors  so  small  that  their  squares,  prod- 
ucts, and  higher  powers  may  be  neglected,  Taylor's  series  gives 


LAWS    OF    ERROR.  31 

This  equation  may  be  said  to  express  the  resultant  actual  error 
x)f  the  function  in  terms  of  the  component  actual  errors,  since 
the  actual  value  of  JQ  is  known  when  the  actual  errors  of 
X,  y,  ,c  ,  .  .  are  known.  It  should  be  carefully  noted  that  the 
quantities  x,  y,  s  .  ..  are  supposed  subject  to  errors  which  are 
independent  of  one  another.  The  discovery  of  the  independent 
sources  of  error  is  sometimes  a  matter  of  difficulty,  and  in  general 
requires  close  attention  on  the  part  of  the  student  if  he  would 
avoid  blunders  and  misconceptions.  Every  investigator  in  work 
of  precision  should  have  a  clear  notion  of  the  error-equation  of 
the  type  (i)  appertaining  to  his  work;  for  it  is  thus  only  that 
he  can  distinguish  between  the  important  and  unimportant 
sources  of  error. 

Prob.  15.  Write  out  the  error-ecjuation  in  accordance  with  (i) 
for  the  function  Q  -  xyz  -\-  x"  log  (;'A). 

Prob.  16.  In  a  plane  triangle  a/b  =  sin  A/^\\\  B.  Find  the  error 
in  (?  due  to  errors  in  /',  .-V,  and  B. 

Prob.  17.   Suppose  in  place  of   the  data  of  problem    i6  that   the 
angles  used  in  computation  are  given  by  the  following  equations  : 
^  =  /?, +  1(180°-^  -  B-  C,),  ^  =  ^,  +  Mi8o°  -A-B^  -  CX 
where  ^,,  B,  ,  C,  are  observed  vahies.     What  then  is  Ja? 

Prob.  18.  If  7C'  denote  the  weight  of  a  body  and  r  the  radius  of 
the  earth,  show  that  for  small  changes  in  altitude,  Aw/7V'—  —  2Ar/r\ 
whence,  if  a  precision  of  one  part  in  500000000  is  attainal)Ie  in  com- 
paring two  nearly  equal  masses,  the  effect  of  a  difference  in  altitude 
of  one  centimeter  in  the  scale-pans  of  a  balance  will  be  noticeable.* 

Art.  10.     Laws  of  Error. 

A  law  of  error  is  a  function  which  expresses  the  relative 
frequency  of  occurrence  of  errors  in  term.s  of  their  magnitudes. 
Thus,  using  the    customary  notation,  let  e  denote  the  magni- 

*  This  problem  arose  with  the  International  Bureau  of  Weights  anti  Measures, 
whose  \vnri<  of  intercomparison  of  the  Prototype  Kilogrammes  attained  a  pre- 
cision indicated  by  a  probable  error  ol  1/500  000  000th  part  of  a  kilogramme. 


32 


PROBABILITY    AND     THEORY    OF    ERRORS. 


tude  of  any  error  in  a  system  of  possible  errors.  Then  the  law 
of  such  s)stem  may  be  expressed  by  an  equation  of  the  form 

J/  =0(6).  (I) 

Representing  e  as  abscissa  and  j  as  ordinate,  this  equation 
gives  a  curve  called  the  curve  of  frequency,  tiie  nature  of  which, 
as  is  evident,  depends  on  the  form  of  the  function  0.  Tiiis 
equation  gives  the  relativ'e  frequency  of  occurrence  of  errors  in 
the  s)'stem  ;  so  that  if  e  is  continuous  the  probability  of  the 
occurrence  of  any  particular  error  is  expressed  by  ju/e  =  0(e)</e; 
which  is  infinitesimal,  as  it  plainly  should  be,  since  in  any  con- 
tinuous system  the  number  of  different  values  of  e  is  infinite. 

Consider  the  simplest  form  of  0(6),  namely,  that  in  which 
0(^e)  =  r,  a  constant.  This  form  of  (p{e)  obtains  in  the  case  of 
tlie  errors  of  tabular  logarithms,  natural  trigonometric  func- 
tions, etc.  In  this  case  all  errors  between  minus  a  half-unit 
and  plus  a  half-unit  of  the  last  tabular  place  are  equally  likely 
to  occur.  Suppose,  to  cover  the  class  of  cases  to  which  that 
just  cited  belongs,  all  errors  between  the  limits  —a  and  -\- a 
are  equally  likely  to  occur.  The  probability  of  any  individual 
error  will  then  be  cp{e)de  =^  cde,  and  the  sum  of  all  such  prob- 
abilities, by  equation  (2),  Art.  4,  must  be  unity.     That  is, 

+a  -fa 

/  <p{e)de  —  c  I  de  —  \.  (2) 

-a  -a 

This  gives  c  =  i/2a,  or  by  (i)  j/  =  1/2^?.     The  curve  of  fre- 
Q  quency  in  this  case  is  shown  in  the  figure, 

D  AB  being  the  axis  of  e  and  OQ  that  of  j. 
It  is  evident  from  this  diagram  that  if  the 
errors  of  the  system  be  considered  with 
respect  to  magnitude  onh',  half  of  them 
should  be  greater  and  half  less  than  a/2. 
This  is  easily  found  to  be  so  in  the  case  of 
tabular  logarithms,  etc. 

As  a  second  illustration  of  (i),  suppose  j  and  e  connected 
by  the  relation^  =  c  Vd"  —  e\  where  a  is  the  radius  of  a  circle, 


TYPICAL   ERRORS    OF    A    SYSTEM.  Xi 

c  a  constant,  and  e  may  have  any  value  between  —  a  and  -j-  a. 
Then  the  condition 

+  a 

cj  de  Vd'  -  e'  =  i 

gives  c  =  2/{a^Ti).  In  this,  as  in  the  preceding  case,  0(-|-  e)  — 
0(—  e),  the  meaning  of  which  is  that  positive  and  negative 
errors  of  the  same  magnitude  are  equally  likely  to  occur.  li 
will  be  noticed,  however,  that  in  the  latter  case  small  errors 
have  a  much  higher  probability  than  those  near  the  limit  a, 
Avhile  in  the  former  case  all  errors  have  the  same  probability. 

In  general,  when  e  is  continuous  0(e)  must  satisfy  the  condi- 
tion   /  (p{e)de  =  I,  the  limits  being  such  as  to  cover  the  entire 

range  of  values  of  e.  The  cases  most  commonly  met  with  are 
those  in  which  0(e)  is  an  even  function,  or  those  in  which 
0(-|-  e)  =  0(—  e).  In  such  cases,  if  ±  a  denote  the  limiting 
value  of  e, 

+  a  a 

fcp{e)de  =  2f<p{e)de  =  I.  (3) 

Art.  11.    Typical  Errors  of  a  System. 

Certain  typical  errors  of  a  system  have  received  special 
desienations  and  are  of  constant  use  in  the  literature  of  the 
theory  of  errors.  These  special  errors  are  the  probable  error, 
the  mean  error,  and  the  average  error.  The  first  is  that  error 
of  the  system  of  errors  which  is  as  likely  to  be  exceeded  as 
not  ;  the  second  is  the  square  root  of  the  mean  of  the  squares 
of  all  the  errors  ;  and  the  third  is  the  mean  of  all  the  errors 
regardless  of  their  signs.  Confining  attention  to  systems  in 
wliich  positive  and  negative  errors  of  the  same  magnitude 
are  equally  probable,  these  typical  errors  are  defined  mathe- 
matically as  follows.     Let 

e^  =  the  probable  error, 

e,„=  the  mean  error, 

Ca  =■  the  average  error. 


34  PROBABILITY  AND  THEORY  OF  ERRORS. 

Then,  observing  (2),  of  Art.  10, 

f(p{6)de  =J(p{e)de  =  f  (p{e)de  =J'(P{e)de  = 

-a  -fp  0  +e^ 

+  a  +a 

3 


y    (0 


=  /  (p(^e)6-de,     e^,  =  2  /  cp{e)ede. 


-a  0 

The  student  should  seek  to  avoid  the  very  common  misap- 
prehension of  the  meaning  of  the  probable  error.  It  is  not 
"  the  most  probable  error,'"  nor  "  the  most  probable  value  of 
the  actual  error";  but  it  is  that  error  which,  disregarding  signs, 
would  occupy  the  middle  place  if  all  tlie  errors  of  the  system 
were  arranged  in  order  of  magnitude.  A  few  illustrations  will 
sufifice  to  fix  the  ideas  as  to  the  typical  errors.  Thus,  take  the 
simple  case  wherein  0(e)  =  ^  =  \ /2a,  which  applies  to  tabular 
logarithms,  etc.      Equations  (i)  give  at  once 

1  ,      ft      l/~  I 

ep=  ±  -  a,     €,„  =  ±  -  '^  3  ,     6^=  ±  ~a. 

2  3  2 

For  the  case  of  tabular  values,  a  =  0.5  in  units  of  the  last 
tabular  place.      Hence  for  such  values 

€p=  ±  0.25,     e„,  =  ±  0.29,     e,  =  ±  0.25. 

Prob.  19.  Find  the  typical  errors  for  the  cases  in  which  the  la\v 
of  error  is  0(e)  =  i:Vd'  —  e\  (pie)  =  c(±a^£),  <p{e)=c  cos" in: £/2ay, 
c  being  a  constant  to  be  determined  in  each  case  and  e  having  any 
value  between  —  a  and  +  a. 

Art.  12.     Laws  of  Resultant  Error. 

When  several  independent  sources  of  error  conspire  to  pro- 
duce a  resultant  error,  as  specified  by  equation  (i)  of  Art.  9, 
there  is  presented  the  problem  of  determining  the  law  of  the 
resultant  error  by  means  of  the  laws  of  the  component  errors. 
The  algebraic  statement  of  this  problem  is  obtained  as  follows 
for  the  case  of  continuous  errors  : 

In  the  equation  (i),  Art.  9,  write  for  brevity 


LAWS    OF    KESULTANT    ERROR.  35 

and  let  the  laws  of  error  of  e,  e^,  e^,  ...  be  denoted  by  (p{e), 

^,(6-,),  0,(^2)  •  •  •     Tlien  the  value  of  e  is  given  by 

e=^, +  ^, +•••  (i) 

The  probabilities  of  the  occurrence  of  any  particular  values 
of  e,,  e„  .  .  .  are  given  by  (/),{£, )de^,  0,(ej^/e,,  .  .  .  ;  and  the 
probability  of  their  concurrence  is  the  probabihty  of  the  cor- 
responding value  of  £.  But  since  tiiis  value  may  arise  in  an 
infinite  number  of  ways  through  tiic  variations  of  e,,  e„  .  .  . 
over  their  ranges,  the  probability  of  e,  or  0(e)c/e,  will  be 
expressed  by  the  integral  of  (p^{e^)d£^(p^{£^)d£^  .  .  .  subject  to 
the  restriction  (i).  Tliis  latter  gives  e,  =  e —  e„  —  £^  .  .  .,  and 
</e,  =  (/£  for  the  multiple  integration  with  respect  to  e,,  e„  .  .  . 
Hence  there  results 

cp{e)d6  =  deJ"(PXe  -  e,  -  ^3  —  •  •  •) 0.(^2)^^2  •  •  •  , 
or 

0(^)  =  y0:(e  -e,  -  e,  -  .  .  .)(p,{e,)d£,J(J^,{£,)d£,  ...       (2) 

It  is  readily  seen  that  this  formula  will  increase  rapidly  in 
complexity  with  the  number  of  independent  sources  of  error.* 
For  some  of  the  most  important  practical  applications,  how- 
ever, it  suf^ces  to  limit  equation  (2)  to  the  case  of  two  inde- 
pendent sources  of  error,  each  of  constant  probability  within 
assigned  limits.  Thus,  to  consider  this  case,  let  e^  var\^  over 
the  range  —  a  to  -{-  a,  and  e,  vary  over  the  range  —  b  to  -\-  b. 
Then  by  equation  (2),  Art.  10, 

0,(e,)  =  i/{2a),      0,(6„)  =  l/{2b). 
Hence  equation  (2)  becomes 


^^^)  =  ifbf'^'^ 


4ab, 

In  evaluating  this  integral  e^  miust  not  surpass  ±  ^  and 
e,  =  e  —  e„  must  not  surpass  ±  (^-  Assuming  a  >  b,  the  limits 
of  the  integral  for  any  value  of  e  =  e^  -\-  e^  b'''\?  between 
—  {a  -j-  b)  and  —  {a  —  b)  are  —  b  and  -}-  (e  -}-  c?).     This  fact  is 

*  The  reader  desirous  of  pursuing  this  phase  of  the  subject  should  consult 
Bessel's  Untersuchungen  ueber  die  VVahrscheiiilichkeit  der  Beobachtungsfehler; 
Abhandlungen  von  Bessei  (Leipzig,  1876),  Vol.  II. 


3G  PROBABILITY    AND    THEORY    OF    ERRORS. 

made  plain  by  a  numerical  example.  For  instance,  suppose 
^  =  5  and  /^  =  3.  Then  —  [a-]-  d)  =  —8  and  —  {a  ~  d)  =  —  3. 
Take  e  =  —  6,  a  number  intermediate  to  —  8  and  —  3.  Then 
the  following  are  the  possible  integer  values  of  e,  and  e,  which 
will  produce  e  =  —  6: 

e  e,       e,  limits  of  e^ 

—  6=  -5  -I,  -i=^{e-\-a), 
=  -  4  -  2, 

=  -  3-  3,  -3^  -b. 

Similarly,  the  limits  of  e^  for  values  of  e  lying  between 
—  {a  —  b)  and  -\- {a  —  b)  are  —  b  and  -|-  h\  and  the  limits  of 
e,  for  values  of  e  between  -{-  {a  —  b)  and  -{-{a-{-  b)  are  +  (e  —  ^) 
and  "1-  b.     Hence 

e-\-a-\-b 


4ab 
2b 


for  —{a-]-b)  <  e<  —  (a—b), 


(3) 


for  _  (^  -  ^)<  e  <  +  (^  -  b), 


'-^-^  for  -^{a-b)<e<+{a^b). 


Thus  it  appears  that  in  this  case  the  graph  of  the  resultant 

law  of  error  is  represented  by  the  upper  base  and  the  two  sides 

of   a  trapezoid,  the  lower  base  beingr  the 
IIOII  ^  .... 

axis  of  6  and  the  line  loining  the  middle 
liioiii  .  .         , 

ponits  of  the  bases  benig  the  axis  of  0(e). 

(See  the  first  figure  in  Art.  13.)     Tlie  prop- 
erties of  (t,),  including  the  determination 

III  .     . 

of  the  limits,  are   also  illustrated    by  the 

nil.  .        , 

adjacent  trapezoid  of    numerals   arranged 

to  represent  the  case  wherein  a  =  0.5  and 
^  z=  0.3.  The  vertical  scale,  or  that  for  cp(e),  does  not,  how- 
ever, conform  exactly  to  that  for  e. 


Ill  ion 

II 1 1  ion 

II 1 1  lion 

II  iiiiion 

III  I  I  II lOI I 


ERRORS  OF  INTERPOLATED  VALUES.  37 

Prob.  20.  Prove  that  the  values  of  0(e)  as  given  by  equation  (3) 
satisfy  the  condition  specified  in  eciuation  (3),  Art.  10, 

Prob.  21.  Examine  equations  (3)  for  the  cases  wherein  a  =  /"and 
<^  =  o;  and  interpret  for  the  latter  case  the  hrst  and  last  of  (3). 

Prob.  22.  Find  from  (3),  and  (i)  of  Art.  11,  the  probable  error  of 
the  sum  of  two  tabular  logarithms. 

Art.  13.     Errors  of  Interpolated  Values. 

Case  I. — One  of  the  most  instructive  cases  to  which  formulas 
(3)  of  Art,  12  are  applicable  is  that  of  interpolated  logarithms, 
trigonometric  functions,  etc.,  dependent  on  first  differences. 
Thus,  suppose  that  %\  and  7',  are  two  tabular  logarithms,  and 
that  it  is  required  to  get  a  value  v  lying  /  tenths  of  the  interval 
from  T',  towards  v.-^.      Evidently 

y   ^  y^  _j_  (,;^    _    2,J  /  =    (l    _    t)V^    +  tV,   ; 

and   hence  if  e,  e^,  e^  denote  the  actual  errors  of  v,  v^,  i\  ,  re- 
spectively, 

^  =  (1  -0^'.  +  ^^.-  (0 

It  is  to  be  carefully  noted  here  that  ^  as  given  by  (i)  re- 
quires the  retention  in  v  of  at  least  one  decimal  place  be- 
yond the  last  tabular  place.  For  example,  let  z'  =  log  (24373) 
from  a  5-place  table.  Then  7\  =  4.38686,  7'.,  =  4.38703, 
v^  —  7\  —  -|- 0.00017,  /  =  0.3,  and  7>  =4.38691.1.  Likewise,  as 
found  from  a  7-place  table,  ^,  =  — 0.45,  e.,  =  -f- 0.37  in  units  of 
the  fifth  place;  and  hence  by  (i)  ^=  —0.20.  That  is,  the 
actual  error  of  ?'  =  4.38691.1  is  =  0.20,  and  this  is  verified  by 
reference  to  a  7-place  table. 

The  reader  is  also  cautioned  against  mistaking  the  species 
of  interpolated  values  here  considered  for  the  species  common- 
ly used  by  computers,  namely,  that  in  which  the  interpolated 
value  is  rounded  to  the  nearest  unit  of  the  last  tabular  place. 
The  latter  species  is  discussed  under  Case  II  below. 

Confining  attention  now  to  the  class  of  errors  specified  by 

equation    (i),  there    result    in  the  notation  of    the  preceding 

article 

e,  =  ( I  —  t)e^,     e„  =  te^,     and     e  —  e  —  e^-\-e.,\ 

and  since  e^  and  e^  each  vary  continuously  between  the  limits 


38 


PROBABILITY  AND  THEORY  OF  ERRORS. 


±0.5  of  a  unit   of  the  last  tabular  place,  a  and  b  in  equations 
(3)  of  that  article  have  the  values 

a  —  0.5(1  —  /),     b  —  0.5/. 

Hence  the  law  of  error  of  the  interpolated  values  is   ex- 
pressed as  follows : 

0(e)  =  -^^^C —  for  values  of  e  betw.  — 0.5  and  —(0.5—/), 
(I        t)t 

= for  values  of  e  betw.  —(0.5—/)  and  -[-(0.5—/),  )■  (2) 


0.5 


for  values  of  e  betw.  -l-(o.5— /)  and  -(-0.5. 


-(I  -/)/ 

The  graph  of  0(e)  for  /  =  1/3  is  shown  by  the  trapezoid 
AB,  BC,  CD  in  the  figure  on  page  40.  Evidently  the  equa- 
tions (2)  are  in  general  represented  by  a  trapezoid,  which  degen- 
erates to  an  isosceles  triangle  when  /  =  1/2. 

The  probable,  mean,  and  average  errors  of  an  interpolated 
value  of  the  kind  in  question  are  readily  found  from  (2),  and 
from  equations  (i)  of  Art.  1 1,  to  be 


=  (i/4)(i-0 

=  1/2  -  {\/2)^2t{\  —  t) 
=  1/4^ 

_  (I  -(I  -2/y)v^ 
-  I  96(1  -  /)/i   • 

I  -  (I  -  2ty 


for  o  <  /  <  1/3, 
for  1/3  </  <  2/3, 
for  2/3  </  <  1. 


}-       (3) 


e,   = 


24(1  -/)/ 

I  -(2/-    l)  = 


for      o  <  /  <  1/2, 


for  1/2  <  /  <  I. 


24(1  -/y 

It  is  thus  seen  that  the  probable  error  of  the  interpolated 
value  here  considered  decreases  from  c.25  to  0.15  of  a  unit  of 
the  last  tabular  place  as  /  increases  from  o  to  0.5,  Hence  such 
values  are  more  precise  than  tabular  values  ;  and  the  computer 
who  desires  to  secure  the  highest  attainable  precision  with  a 
given  table  of  logarithms  should  retain  -^ne  additional  figure 
bej'oiid  the  last  tabular  place  in  interpolated  values. 


ERRORS  OF  INIERI'OLATED  VALUES  39 

Case  II. — ReCLirrinj^  to  the  ccjuatioii  7'  =  7',  -f-  t{v.^  —  t',)  for  an 
interpolated  value  v  in  terms  of  two  consecutive  t.ibular  values 
7',  and  7'j,  it  will  be  observed  that  if  the  quantity  t{v^—  7',)  is 
rounded  to  the  nearest  unit  of  the  last  tabular  place,  a  new  error 
is  introduced.  For  example,  if  z'^  =  log  1633  =  3.21299,  and 
^2  =  log  1^34  =  3-21325  from  a  5-place  table,  v^  —  v^  =  -f-  26 
units  of  the  last  tabular  place  ;  and  if  /  =  1/3,  /(7'.^  —  v^)  =  8^  ; 
so  that  by  the  method  of  interpolation  in  question  there  results 
V  =  3.21299  +9  =  3-21308.  Now  the  actual  errors  of  7',  and 
7'j  are,  as  found  from  a  7-place  table,  —  0.38  and  -|-0  2i  in  units 
of  the  fifth  place.  Hence  the  actual  error  of  7'  is  by  equation 
(0»  I  X  —  0.38  -f  J  X  +  0.21  —  1  =  —  0.52,  as  is  shown  di- 
rectly by  a  7-place  table. 

It  appears,  then,  that  in  this  case  the  error-equation  cor- 
responding to  (i)  is 

e=  {i  -  t)c,  +  te,  +  e, ,  (4) 

wherein  ^,  and  i\  are  the  same  as  in  (i)  and  ^'3  is  the  actual  error 
that  comes  from  rounding  t{v^  —  7',)  to  the  nearest  unit  of  the 
last  tabular  place. 

The  error  e^,  however,  difTers  radically  in  kind  from  r,  and 
e^.  The  two  latter  are  continuous,  that  is,  they  may  each  have 
any  value,  between  the  limits  —  0.5  and  +0-5  \  while  ^3  is  dis- 
continuous, being  limited  to  a  finite  number  of  values  depend- 
ent on  the  interpolating  factor  t.  Thus,  for  /  =  1/2  the  only 
possible  values  of  c^  are  O  -\-  1/2,  and  —  1/2  ;  likewise  for  /  — 
1/3,  the  only  possible  values  of  ^3  are  o,  -j-  1/3,  and  —  1/3.  It 
is  also  clear  that  the  maximum  value  of  t\  which  is  constant  and 
equal  to  1/2  for  (i),  is  variable  for  (4)  in  a  manner  dependent 
on  /.     For  example,  in  (4), 

The  maximum  of  ^  =  1/2  +  r/2  =  l,       for  /  =  1/2, 

"  ^  =  1/2  +  1/3  =  5/6,  "    /'=i/3. 

«  "  "^=1/2  +  1/2=1,       "    /  =  1/4, 

'V=:    1/2  4-   2/5    =9/10"      /  ~    l/5- 

The  determination  of  the  law  of  error  for  this  case  presents 
some  novelty,  since  it  is  essential  to  combine  the  continuous 
errors  (i  —  t)i\  and  te^  with  the  discontinuous  error  e^.     The 


40 


PROBABILITY  AND  THEORY  OF  ERRORS, 


simplest  mode  of  attacking  the  problem  seems  to  be  the  fol- 
lowing quasi-geometrical  one.  In  the  notation  of  Arts.  12  and 
13,  put  in  (4)  ^  =  e,  (i  —  /)t\  =  e,,  /r,  =  e^,  and  c^  =  e^.     Then 

e  =  (e,  -f  ej  +  63.  (5) 

The  law  of  error  for  (e^  -|-  e^)  is  given  by  equation  (2)  for  any 
value  of  A  Hence  for  a  given  value  of  t  there  will  be  as  many 
expressions  of  0(e)  as  there  are  different  values  of  e,.  The 
graphs  of  0(e)  will  all  be  of  the  same  form  but  will  be  differently 
placed  with  reference  to  the  axis  of  0(e).     Thus,  if  /"  j=  1/3  the 

values  of  e.^  are  —  1/3,  o,  and 
-|-  1/3,  and  these  are  equally 
likely  to  occur.  For  e^  =  o  the 
graph  is  given  directly  by  (2), 
and  is  the  trapezoid  A  BCD 
symmetrical  with  respect  to  OQ. 
For  £^  =  —  1/3  the  graph  is 
abQd,  of  the  same  form  as 
ABCD  but  shifted  to  the  left 
by  the  amount  of  e-g  =  —  1/3. 
Similarly,  the  graph  for  the  case 
of  £3  =  -|-  1/3  is  a'Qb'd\  and  is  produced  by  shifting  ABCD  to 
the  right  by  an  amount  equal  to  -|-  1/3. 

Now,  since  the  three  systems  of  errors  for  this  case  are 
equally  likely  to  occur,  they  may  be  combined  into  one  system 
by  simple  addition  of  the  corresponding  element  areas  of  the 
several  graphs.  Inspection  of  the  diagram  shows*  that  the 
resultant  law  of  error  is  expressed  by 

0(e)  =  (i/4)(5  +  6e)     for  -  5/6  <  e  <  _  1/6,  >| 

=  1  for  -  1/6  <  e  <  +  1/6,  j-         (6) 

=  (i/4)(5  -  6e)      for  +  I /6  <  e  <  4-  5/6.  J 

This  is  represented  by  a  trapezoid  whose  lower  base  is  10/6, 
upper  base  2/6,  and  altitude  i. 


*  Sum  the  three  areas  and    divide    by  3  to  make    resultant    area  =  i,    as 
required  by  equation  (3),  Art.  10. 


ERRORS  OF  INTERPOLATED  VALUES. 


41 


As  a  second  illustration,  consider  equation  (5)  for  the  case 
/=  1/2.     In  this  case  e,  must  be  either  o  or   1/2,  the  sign  of 
which  latter  is  arbitrary.      For  e^  =  o,  equations  (2)  give 
0(e)  =  2  -|-  46     for      _  1/2  <  6  <  O,  I 

=  2  —  4e     for     o  <  e  <-|-  1/2.  \  (7) 

This   function    is   represented  by  the   isosceles  triangle  AQE 

whose  altitude  OQ  is  twice  the  base  AE. 

Similarly  0(e)  for  e,  =  -|-  1/2  would 

be  represented  by  the  tna.ng\e  AQE  dis- 
placed to  the  right  a  distance  1/2  ;  and 
if  the  two  systems  for  ^3=0  and  63  = 
-|-  1/2  be  combined  into  one  system, 
their  resultant  law  of  error  is  evidently 
0(e)=  i-|-2e     for— i/2<e<o,      \ 

=  I  for  o<  e<  +  i/2,       t      (8) 

=  2  —  2e  for  -f-  1/2  <  e<  I  ;   ) 
the  graph  of  which  is  ABCD.    On  the 
other  hand,  if  the  errors  in  this  combined  system  be  considered 
with  respect  to  magnitude  only,  the  law  of  error  is 

0(e)  =  2(1  -e)      for     o  <  e<  I,  (9) 

the  graph  of  which  is  OQD. 

The  student  should  observe  that  (6),  (7),  (8),  and  (9)  satisfy 

the    condition    /  (p{e)d£  =  i    if  the  integration  embraces   the 

whole  range  of  e. 

Tiie  deteimination  of  the  general  form  of  0(e)  in  terms  of 
the  interpolating  factor  /  for  the  present  case  presents  some 
dif^culties,  and  there  does  not  appear  to  be  any  published  solu- 
tion of  this  problem.*  The  results  arising  from  one  phase  of 
the  problem  have  been  given,  however,  by  the  author  in  the 
Annals  of  Mathematics, f  and  may  be  here  stated  without 
proof.  The  phase  in  question  is  that  wherein  /  is  of  the  form 
1//1,  n  being  an}'  positive  integer  less  than  twice  the  greatest 


*  The  author  explained  a  general  method  of  solution  in  a  paper  read  at  the 
summer  meeting  of  the  American  Mathematical  Society,  August,  1895. 
f  Vol.  II,  pp.  54-59. 


42  PROBABILITY    AND    THEORY    OF    ERRORS. 

tabular  difference  of  the  table  to  which  the  formulas  are  ap- 
plied. For  this  restricted  form  of  /  the  possible  maximum 
value  of  e  as  given  by  equation  (5)  is,  in  units  of  the  last 
tabular  place,  {2n  —  i)/;/  for  n  odd  and  i  for  h  even. 

The  possible  values  of  e^  of  equation  (5)  are 
O, 


I 

2 

71  —    I 

2n 

for  71  odd, 

I 

±  -.. 

71 

2 

71  —  2 

•  •  •   ^        _         » 

211 

± 

I 

2 

for  71  even. 

o, 

An  important  fact  with  regard  to  the  error  1/2  for  ;/  even 
is  that  its  sign  is  arbitrary,  or  is  not  fixed  by  the  computation 
as  is  the  case  with  all  the  other  errors.  However,  the  com- 
puter's rule,  which  makes  the  rounded  last  figure  of  an  inter- 
polated value  even  when  half  a  unit  is  to  be  disposed  of,  will, 
in  the  long-run,  make  this  error  as  often  plus  as  minus. 

The  laws  of  error  which  result  are  then  as  follows : 

For  n  odd. 

0(6)  =  I  for  6  between  —  1/2;/  and  -\-  i/2«, 

71       (271  —    I  \ 

0(e)  = I ±  e    for  ebetw.  =Fi/2«  and  ^{27i—\)/27i. 

^  71  —    \\       271  I  ' 

For  n  even. 

n        (271  —  2         \ 
(p(€)  =  — ,   ±  e    for  e  between  o  and  =F  i/«, 

^    '  2{7l  —    \)\        71  I  T^      /      ' 

;/     (271  —  I  \  , 

=  -I— ±  e)  for  e  betw.  =F  ^ "■  and  T  {71  —  \)/7t, 

71  —    l^       271  I  ji     y 

71 

=  —. r(i  +  e)  for  e  between  T  {n—  i/«)  and  =F  i. 

2{7l  —   If  ^  \  I     I 

By  means  of  these  formulas  and  (i)  of  Art.  1 1  the  probable, 
mean,  and  average  errors  for  any  value  of  ;/  can  be  readily 
found.  The  following  table  contains  the  results  of  such  a  com- 
putation for  values  of  ;/  ranging  from  i  to  10.  The  maximum 
actual  error  for  each  value  of  71  is  also  added.  The  verifica- 
tion of  the  tabular  quantities  will  afford  a  useful  exercise  to  the 
student. 


ERRORS  OF  INTERPOLA'l  ED  VALUES.  43 

Typical  Errors   of  Intkrpolated  Logarithms,  etc. 


Interpotaling 
Factor, 

Probable  Error. 

Mean  Error. 

Average  Error. 

Maximum 

t=  i/n 

/^ 

m 

'^a 

Actual  Error. 

I 

0.250 

0.289 

0.250 

1/2 

1/2 

.292 

.408 

•333 

I 

1/3 

.256 

•347 

.287 

5/6 

>/4 

.276 

.382 

■3'3 

I 

•/5 

.268 

.370 

•303 

9/10 

1/6 

277 

•385 

•3'5 

I 

1/7 

.274 

.380 

•3" 

13/14 

1/3 

.279 

•3S9 

.3>8 

I 

1/9 

.278 

.3S6 

•  3'6 

17/18 

i/io 

.281 

•392 

.320 

I 

When  the  interpolating  factor  /  has  the  more  general  forrn 
J/1//1,  wherein  w  and  ft  are  integers  with  no  common  factor,  the 
possible  values  of  e^  are  the  same  as  for  t  =  i/n.  But  equa- 
tions (3)  of  Art.  12  are  not  the  same  for  /  =  vi///  as  for/  =  i/;/, 
and  hence  for  the  more  general  form  of  /,  0(e)  assumes  a  new 
type  which  is  somewhat  more  complex  than  that  discussed 
above.  The  limits  of  this  work  render  it  impossible  to  extend 
the  investigation  to  these  more  complex  forms  of  0(e).  It  may 
suf^ce,  therefore,  to  give  a  single  instance  of  such  a  function, 
namely,  that  for  which  /  =  2/5.     For  this  case 

0(e)  =  I  for  e  between  o  and  =F  i/io, 

=  (5/6)(i3/iO  ±  e)  for  e  between  ^  i/io  and  q:  3/10, 
=  (5/3)(4/5  ±  e)       for  e  between  T  3/10  ''^'icl  T  7/10, 
=  (5/6)(9/iO±  e)     for  e  between  =F  7/10  and  ^p  9/10. 
The   graph  of  the  right-hand  half  of    ^    b 
this  function  is  shown  in  the  accompany- 
ing   diagram,    the    whole    graph    being 
symmetrical  with   respect  to  OA,  or  the 
axis  of  0(e). 

Attention  maybe  called  to  the  strik- 
ing resemblance  of  this  graph  to  that  of 
the  law  of  error  of  least  squares. 


Prob.  23.   Show   from   equations   (3)  that   e,„  varies  from  i/r  12 

=  0.29  — ,  for  /  =  o,  to  i/v  24  =  0.20  +,  for  /  =  0.5  ;  and  that  e^ 
varies  from  0.25  to  1/6  for  the  same  limits. 


44  PROBABILITY  AND  THEORY  OF  ERRORS. 

Prob.  24.  Show  that  the  probable,  mean,  and  average  errors 
for  the  case  of  /=  2/5  cited  above  (p.  43)  are  ±  0.261,  ±  0.251, 
and  ±  0.290,  respectively. 

Art.  14.  Statistical  Test  of  Theory. 
A  statistical  test  of  the  theory  developed  in  Art.  13  may 
be  readily  drawn  from  any  considerable  number  of  actual  er- 
rors of  interpolated  values  dependent  on  the  same  interpolating 
factor.  The  application  of  such  a  test,  if  carried  out  fully  by 
the  student,  will  go  far  also  towards  fixing  clear  notions  as  to 
the  meaning  of  the  critical  errors. 

Consider  first  the  case  in  which  an  interpolated  value  falls 
midway  between  two  consecutive  values,  and  suppose  this 
interpolated  value  retains  two  additional  figures  beyond  the 
last  tabular  place.  Then  by  equations  (2),  Art.  13,  the  law  of 
error  of  this  interpolated  value  is 

0(e)  =  2  -(-  4e  for  e  between  —  0.5  and  o 
=  2  —  4e  for  e  between  o  and  -f-  0.5. 

Hence  by  equation  (i)  of  Art.  1 1,  or  equation  (3)  of  Art.  12,  the 
probable  error  in  this  system  of  errors  is  ^  —  {^)  V2  :=  o.i^. 
It  follows,  therefore,  that  in  any  large  number  of  actual  errors 
of  this  system,  half  should  be  less  and  half  greater  than  0.15. 
Similarly,  of  the  whole  number  of  such  errors  the  percentage 
falling  between  the  values  0.0  and  0.2  should  be 

+  0-2  +0.2 

J  (p{e)de  =  2  J  (2  —  ^e)de  =  0.64  ; 

-0.2  0 

that  is,  sixty-four  per  cent  of  the  errors  in  question  should  be 
less  numerically  than  0.2. 

To  afTord  a  more  detailed  comparison  in  this  case,  the  act- 
ual errors  of  five  hundred  interpolated  values  from  a  5-place 
table  have  been  computed  by  means  of  a  7-place  table.  The 
arguments  used  were  the  following  numbers  :  20005,  20035, 
20065,  20105,  20135,  etc.,  in  the  same  order  to  36635.  The 
actual  and  theoretical  percentages  of  the  whole  number  of 
errors  falling  between  the  limits  0.0  and  o.  i,  o.i  and  0.2,  etc., 
are  shown  in  the  tabular  form  following:: 


STATISTICAL    TEST    OF    THEORY.  45 

Limits  of  Errors.  „  A^'"^'  Theoretical 

Percentage.  Percentage. 

O.oaiido.  I 33.2  36 

O.  I  and  0.2 30.2  28 

0.2  and  0.3 19.0  20 

0.3  and  0.4 13.2  12 

0.4  and  0.5 4.4  4 

0.0  and  O.I  5 51.4  50 

The  agreement  shown  here  between  the  actual  and  theoretical 
percentages  is  quite  close,  tiie  maximum  discrepancy  being  2.8 
and  the  average  1.5  per  cent. 

Secondly,  consider  the  case  of  interpolated  mid-values  of  the 
species  treated  under  Case  II  of  Art.  13.  The  law  of  error  for 
this  case  is  given  by  the  single  equation  (9)  of  Art.  13,  namel)', 
0(e)  =  2(1  —  e),  no  regard  being  paid  to  the  signs  of  the  errors. 
The  probable  error  is  then  found  from 


« 


whence  6p  =  i  —  -^  V2  =  0.29.  Similarly,  the  percentage  of 
the  whole  number  of  errors  which  may  be  expected  to  lie,  for 
example,  between  0.0  and  0.2  in  this  system  is 


0  2 


2  /  (i  —  e)(/6  =  0.36. 


0 


Using  the  same  five  hundred  interpolated  values  cited 
above,  but  rounding  them  to  the  nearest  unit  of  the  last  tabu- 
lar place  and  computing  their  actual  errors  by  means  of  a  7-place 
table,  the  following  comparison  is  afforded  : 

...        ,  „  Actual  Thcoreiical 

Limits  of  Errors.  Percentage.  Percentage. 

0.0  and  0.2 35.8  36 

0.2  and  0.4  27.8  28 

0.4  and  0.6  1 8.6  20 

0.6  and  0.8 1 2.2  12 

0.8  and  i.o  5.6  4 

0.0  and  0.29 49.8  50 


46  PROBABILITY  AND  THEORY  OF  ERRORS. 

The  agreement  shown  here  between  the  actual  and  theoretical 
percentages  is  somewhat  closer  than  in  the  preceding  case,  the 
maximum  discrepancy  being  only  1.6  and  the  average  only  0.6 
per  cent. 

Finally,  the  following  data  derived  from  one  thousand  act- 
ual errors  may  be  cited.  The  errors  of  one  hundred  inter- 
polated values  rounded  to  the  nearest  unit  of  the  last  tabular 
place  were  computed  *  for  each  of  the  interpolating  factors 
0.1,  0.2,  .  ,  .  0.9.  The  averages  of  these  several  groups  of  act- 
ual errors  are  given  along  with  the  corresponding  theoretical 
errors  in  the  parallel  columns  below: 

Interpolating  Actual  Theoretical 

Factor.  Average  Error.       Average  Error. 

O.I 0.338  0.320 

0.2 o..?88  0.303 

0.3 0.321  0.304 

0.4 0.268  0.290 

0.5 0.324  0.333 

0.6 ...     0.276  0.290 

0.7 0.32  I  0.304 

0.8 0.289  0.303 

0.9 0.347  0.320 

The  average  discrepancy  between  the  actual  and  theoret- 
ical values  shown  here  is  0.017.  It  is,  perhaps,  somewhat 
smaller  than  should  be  expected,  since  the  computation  of  the 
actual  errors  to  tlnee  places  of  decimals  is  hardly  warranted 
by  the   assumption   of   dependence    on    first    differences  only. 

The  averatfe  of  the  whole  number  of  actual  errors  in  this 
case  is  0.308,  which  agrees  to  the  same  number  of  decimals 
with  the  average  of  the  theoretical  errors,  f 

*  By  Prof.  H.  A.  Howe.  See  Annals  of  Mathematics,  Vol.  Ill,  p.  74. 
The  theoretical  averages  were  furnished  to  Prof.  Howe  by  the  author. 

f  The  reader  who  is  acquainted  with  the  elements  of  the  method  of  least 
squares  will  find  it  instructive  to  apply  that  method  to  equation  (i),  Art.  13, 
and  derive  the  probable  error  of  e.     This  is  frequently  done  without  reserve  by 


STATISTICAL    TEST    OF    THEORY.  47 

Prob.  25.  Apply  formulas  (3)  of  Art.  12  to  the  case  of  the  sum 
or  difference  of  two  tabular  logarithms  and  derive  the  correspond- 
ing values  of  the  probable,  mean,  and  average  errors.  The  graph 
of  4>{e)  is  in  this  case  an  isosceles  triangle  whose  base,  or  axis  of  e, 
is  2,  and  whose  altitude,  or  axis  of  0(e),  is  i. 

those  familiar  with   least  squares.      Thus,  the  probable  error  of  <?i  or  ^2  being 
0.25,  the  probable  error  of  e  is  found  to  be 


0.25  Vl    —  2/  +  2/-. 

This  varies  between  0.25  for  ^  =  o  and  o.iS  for  /  =  5  ;  while  the  true  value  of 
the  probable  error,  as  shown  by  equations  (3),  Art.  13,  varies  from  0.25  to  0.15 
for  the  same  values  of  t.  It  is,  indeed,  remarkable  that  the  method  of  least 
squares,  which  admits  infinite  values  for  the  actual  errors  d  and  e^,  should  give 
so  close  an  approximate  formula  as  the  above  for  the  probable  error  of  c. 

Similarly,  one  accustomed  to  the  method  of  least  squares  would  be  inclined 
to  apply  it  to  equation  (4),  Art.  13,  to  determine  the  probable  error  of  e.  The 
natural  blunder  in  this  case  is  to  consider  ei,  ei ,  and  ^3  independent,  and  e-.,  like 
fi  and  i'i  continuous  betweer  the  limits  0.0  and  0.5  ;  and  to  assign  a  probable 
error  of  0.25  to  each.      In  t'.is  manner  the  value 


0.25  ^2(1  -  i  +  i^) 

is  derived.  But  this  is  absurd,  since  it  gives  0.25  V^2  instead  of  0.25  for  t  =■  o. 
The  formula  fails  then  to  give  even  approximate  results  except  for  values  of  / 
near  0.5. 


INDEX. 


Average  error,  33,  34. 

of   interpolated   logarithm,   38,   43. 
of  tabular  logarithm,  34. 

Babbage: 

Ninth  Bridgewater  treatise  of,  26. 
Bernoulli,  James: 

theorem  of,  22. 

work  cited,  8. 
Bert';t;id,  work  cite,',  '>.^. 
Bessel,  work  cited,  35. 

Chance,  games  of,  7. 
Combinations,  13-16. 

formulas  for,  14-16. 

table  of,  15. 
Concurrent  events,  19-21. 

De  Moivre,  work  cited,  8. 

De  Morgan,  work  cited,  13,  22. 

Error  equation,  31. 
function,  31. 
Errors,  theory  of,  30-47. 

Fermat,  7,  8. 

Games  of  chance,  7. 

Gamma  function,  28. 

Geographical    tables    (of    Smithsonian 

Institution)  cited,  10. 
Graphs  of  laws  of  error,  32,  36,  40,  41, 

43- 

Howe,  computation  of,  cited,  46. 
Huygens,  work  cited,  8. 

Integral,  probability,  23. 
table  of,  24. 

Jevons,  work  cited,  16. 

Laws  of  error,  31-33. 

interpolated  logarithms,  34-47. 

least  squares,  10,  43,  46,  47. 

tabular  logarithms,  32. 
Least  squares,  10,  43,  47. 
Laplace,  work  cited,  9,  22. 
Logarithmic  tables,  37-43. 


Mean  error,  ^;i,  34. 

of  interpolated  logarithms,  38,  43. 

of  tabular  logarithms,  34. 
Mere,  Chevalier  de,  7. 
Method  of  least  squares,   10,  46,  47. 
Montmort,  work  cited  8. 

Observations,  errors  of,  30,  31. 

Pascal,  7,  8. 
Permutations,  11-18. 

,   formulas  for,  11,  12. 
table  of,  II. 
Poisson,  work  cited,  7,  20,  24. 
Probable  error,  ^^,  34. 

of  interpolated  logarithms,  38,  43. 

of  tabular  logarithms,  34. 
Probabilities,  16-30. 

direct,  16-18. 

inverse,  24-27. 

of  concurrent  events,  19-21. 

of   concurrent    testimony,    26. 

of  future  events,  27-30. 
Probability  integral,  23. 

Resultant  error,  34. 

Shortrede,  tables  cited,  13. 
Statistical  test  of  theory,  44-46. 
Stirling's  theorem,  22,  23. 

Table  of  combinations,  15. 

of  permutations,  11. 

of  probability  integral,  24. 

of  statistical  test,  45,  46. 

of  typical  errors,  43. 
Tabular  values,  errors  of,  34-38. 
Theory  of  errors,  30-47. 

of  interpolated  values,  37-46. 
Todhunter,  I.,  work  cited,  7,  28. 
Typical  errors,  $3,  43. 

Values  of  combinations,  15. 
of  permutations,  11, 
of  typical  errors,  43. 


^ 


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